Finite Elements in Vector Lattices 9783110350784, 9783110350777

Table of contents :
Contents
1. Introduction
2. Ordered vector spaces and vector lattices
2.1 Ordered vector spaces and positive operators
2.2 Vector lattices
2.3 Ordered normed spaces
2.4 Normed Riesz spaces and Banach lattices
2.5 Representation of Banach lattices
3. Finite, totally finite and selfmajorizing elements
3.1 Finite and totally finite elements in vector lattices
3.2 Finite elements in Banach lattices
3.3 Finite elements in sublattices and in direct sums of Banach lattices
3.3.1 Finite elements in sublattices
3.3.2 Finite elements in the bidual of Banach lattices
3.3.3 Finite elements in direct sums of Banach lattices
3.4 Selfmajorizing elements in vector lattices
3.4.1 The order ideal of all selfmajorizing elements in a vector lattice
3.4.2 General properties of selfmajorizing elements
3.4.3 Examples of selfmajorizing elements
3.5 Finite elements in l-algebras and in product algebras
3.5.1 Lattice ordered algebras
3.5.2 Finite elements in unitary l-algebras
3.5.3 Finite elements in nonunitary f-algebras
3.5.4 Finite elements in product algebras
4. Finite elements in vector lattices of linear operators
4.1 Some general results
4.2 Finiteness of regular operators on AL-spaces
4.3 Finite rank operators in the vector lattice of regular operators
4.4 Some vector lattices and Banach lattices of operators
4.4.1 Vector lattices of operators
4.4.2 Banach lattices of operators
4.5 Operators as finite elements
4.6 Finite rank operators as finite elements
4.7 Impact of the order structure of V(E, F) on the lattice properties of E and F
5. The space of maximal ideals of a vector lattice
5.1 Representation of vector lattices by means of extended real continuous functions
5.2 Maximal ideals and discrete functionals
5.3 The topology on the space of maximal ideals of a vector lattice
5.4 The Hausdorff property of M
6. Topological characterization of finite elements
6.1 Topological characterization of finite, totally finite and selfmajorizing elements
6.1.1 The canonical map and the conditional representation
6.1.2 Topological characterization of finite elements
6.1.3 Topological characterization of totally finite elements
6.1.4 Topological characterization of selfmajorizing elements
6.2 Relations between the ideals of finite, totally finite and selfmajorizing elements
6.3 The topological spaceMfor vector lattices of type (Σ)
6.4 Examples
7. Representations of vector lattices and their properties
7.1 A classification of representations and the standard map
7.2 Vector lattices of type (Σ) and their representations
8. Vector lattices of continuous functions with finite elements
8.1 Vector lattices of continuous functions with many finite functions
8.2 Finite elements in vector lattices of continuous functions
8.3 An isomorphism result for vector lattices of continuous functions
9. Representations of vector lattices by means of continuous functions
9.1 Representations which contain finite functions
9.2 The existence of Φα- representations for vector lattices of type (Σ)
9.3 LF-vector lattices
9.4 Vector lattices of type (CM)
10. Representations of vector lattices by means of bases of finite elements
10.1 Bases of finite elements and α -representations
10.2 Representations by means of R-bases of finite elements
10.3 Some properties of the realization space
List of Examples
List of Symbols
Bibliography
Index

Citation preview

Martin R. Weber Finite Elements in Vector Lattices

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Journal of Numerical Mathematics Hoppe, Kuznetsov (Eds.-in-Chief) ISSN 1570-2820, e-ISSN 1569-3953

Martin R. Weber

Finite Elements in Vector Lattices |

Mathematics Subject Classification 2010 46B40, 46B42, 46A40, 46E05, 47B65, 06F25 Author Prof. Dr. Martin R. Weber (Seniorprofessor) TU Dresden Institut für Analysis Helmholtzstr 10 01062 Dresden [email protected]

ISBN 978-3-11-035077-7 e-ISBN 978-3-11-035078-4 Set-ISBN 978-3-11-035079-1 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2014 Walter de Gruyter GmbH, Berlin/Boston Typesetting: PTP-Berlin, Protago-TEX-Production GmbH, Berlin Printing and binding: CPI books GmbH, Leck ♾Printed on acid-free paper Printed in Germany www.degruyter.com

| Gewidmet meiner lieben Frau Ute, meinen lieben Kindern Annett und Alexander und meinem verehrten Lehrer Boris Michailowitsch Makarow

Contents 1

Introduction | 1

2 2.1 2.2 2.3 2.4 2.5

Ordered vector spaces and vector lattices | 4 Ordered vector spaces and positive operators | 4 Vector lattices | 6 Ordered normed spaces | 11 Normed Riesz spaces and Banach lattices | 12 Representation of Banach lattices | 16

3 3.1 3.2 3.3

Finite, totally finite and selfmajorizing elements | 18 Finite and totally finite elements in vector lattices | 18 Finite elements in Banach lattices | 29 Finite elements in sublattices and in direct sums of Banach lattices | 33 Finite elements in sublattices | 33 Finite elements in the bidual of Banach lattices | 37 Finite elements in direct sums of Banach lattices | 39 Selfmajorizing elements in vector lattices | 41 The order ideal of all selfmajorizing elements in a vector lattice | 42 General properties of selfmajorizing elements | 44 Examples of selfmajorizing elements | 47 Finite elements in ℓ-algebras and in product algebras | 49 Lattice ordered algebras | 49 Finite elements in unitary ℓ-algebras | 52 Finite elements in nonunitary 𝑓-algebras | 57 Finite elements in product algebras | 63

3.3.1 3.3.2 3.3.3 3.4 3.4.1 3.4.2 3.4.3 3.5 3.5.1 3.5.2 3.5.3 3.5.4 4 4.1 4.2 4.3 4.4 4.4.1 4.4.2 4.5 4.6 4.7

Finite elements in vector lattices of linear operators | 69 Some general results | 70 Finiteness of regular operators on 𝐴𝐿-spaces | 75 Finite rank operators in the vector lattice of regular operators | 77 Some vector lattices and Banach lattices of operators | 81 Vector lattices of operators | 83 Banach lattices of operators | 84 Operators as finite elements | 90 Finite rank operators as finite elements | 92 Impact of the order structure of V(𝐸, 𝐹) on the lattice properties of 𝐸 and 𝐹 | 96

viii | Contents 5 5.1 5.2 5.3 5.4 6 6.1

The space of maximal ideals of a vector lattice | 100 Representation of vector lattices by means of extended real continuous functions | 100 Maximal ideals and discrete functionals | 103 The topology on the space of maximal ideals of a vector lattice | 107 The Hausdorff property of M | 109

6.3 6.4

Topological characterization of finite elements | 115 Topological characterization of finite, totally finite and selfmajorizing elements | 115 The canonical map and the conditional representation | 116 Topological characterization of finite elements | 121 Topological characterization of totally finite elements | 125 Topological characterization of selfmajorizing elements | 129 Relations between the ideals of finite, totally finite and selfmajorizing elements | 131 The topological space M for vector lattices of type (Σ) | 134 Examples | 138

7 7.1 7.2

Representations of vector lattices and their properties | 144 A classification of representations and the standard map | 144 Vector lattices of type (Σ) and their representations | 148

8 8.1

Vector lattices of continuous functions with finite elements | 157 Vector lattices of continuous functions with many finite functions | 157 Finite elements in vector lattices of continuous functions | 162 An isomorphism result for vector lattices of continuous functions | 167

6.1.1 6.1.2 6.1.3 6.1.4 6.2

8.2 8.3

9 9.1 9.2 9.3 9.4 10 10.1

Representations of vector lattices by means of continuous functions | 171 Representations which contain finite functions | 171 The existence of Φ𝛼-representations for vector lattices of type (Σ) | 177 𝐿𝐹-vector lattices | 182 Vector lattices of type (𝐶𝑀 ) | 184 Representations of vector lattices by means of bases of finite elements | 191 Bases of finite elements and 𝛼-representations | 191

Contents

10.2 10.3

|

Representations by means of R-bases of finite elements | 195 Some properties of the realization space | 199

List of Examples | 207 List of Symbols | 209 Bibliography | 211 Index | 217

ix

1 Introduction Since the 1950s, ordered vector spaces, vector lattices and such spaces equipped with an appropriate norm or topology have been studied by many authors. The general theory of ordered, normed ordered vector spaces, vector lattices and normed vector (Banach) lattices is comprehensively treated in the literature. The related main monographs, e. g., [2, 9, 56, 59, 60, 84, 95, 100, 109, 120, 143, 144], provided here in chronological order, contain as a rule the research results which were current at the time of their publishing. In the last forty years the applications of the theory have grown remarkably. This development has been fostered in many branches of mathematics (such as optimization, numerical methods, positive solutions of equations, positive systems, positive semigroups, measure theory etc.) by the manifold aspects summarized under the heading of positivity. It is impossible to provide only a rough survey of the applications spread over many fields of the present-day mathematical research, so we shall only refer only to some of them: – economics, equilibrium theory [11–13]; – convex operators, extremal problems, Choquet theory, variational methods [7, 20, 50]; – positive solutions of operator equations, integral operators, fixed point equations, maximum principles [14, 19, 30, 58, 65]; – positive systems [45, 66]; – semigroups of positive operators [18, 42, 96]; – measure theory [48, 52]; – stochastic processes, martingale theory [72–74]. On the other hand, special problems e. g., cones in Banach spaces [10, 15–17, 106, 121, 122], dominated operators [77], integral operators [30], order continuous norms [139] and miscellaneous others complete the general theory by many new and particular aspects. The subject of investigation in the present book is a class (order ideal) of particular elements in Archimedean vector lattices which originated from and are closely related to continuous functions with compact support on a topological noncompact Hausdorff space. The topic of finite elements in the context of vector lattices appeared in the early 1970s. The first explicit definition dates back to 1972, when B. M. Makarov and the author formulated the vector lattice characterization of such elements in arbitrary Archimedean vector lattices. The fundamental formula in its definition characterizes the interaction of a finite element with all other elements in the vector lattice. This book is the first systematical treatment of the theory of finite elements in Archimedean vector lattices and contains the results known on this topic up to the year 2013. We assembled here all contributions achieved by a number of mathemati-

2 | 1 Introduction cians published in the papers [36–38, 54, 89–93, 114, 124–127, 129, 131, 132]. The author thanks all his coauthors for the cooperative and fruitful collaboration on this new research stream as a part of the theory of vector and Banach lattices. Some early results were published in Russian and German and are sometimes difficult to access, so it should be useful to present the main results summarized in a book issued in English. It is hoped that this book will encourage further studies in the field opened up by the investigation of the concepts of finite, totally finite and selfmajorizing elements in vector lattices. The vector space 𝐶(𝑄) of all real continuous functions defined on a locally compact topological Hausdorff space 𝑄 and its subspace K(𝑄) of all continuous functions with compact support have a very rich structure and interesting properties from several points of view. Moreover, K(𝑄) and other subspaces of 𝐶(𝑄) are also used isomorphically to represent many other abstract spaces. It is convenient to look at the elements of an abstract mathematical object as continuous real-valued functions on a topological space. The reasons and advantages may be twofold: first, continuous realvalued functions are considered prototypes for abstract elements, the nature of which is unknown (and therefore enabling one to discover the general features and properties of abstract ones), where the usual operations defining the structure of the object under investigation reduce to the usual (natural) pointwise algebraic and order operations between continuous functions. Second, spaces of continuous functions have been systematically and thoroughly studied for at least two centuries by many authors (see, e. g., [49, 61, 62, 109, 112, 123]) from different points of view with the result that much is known about them and allowing, therefore, a justified hope that more and deeper properties of the particular structure will be obtained. Therefore, nearly all theories of particular mathematical structures such as algebras, rings, lattices and others are accompanied by a representation theory of such structures mostly by means of continuous functions on some topological space. This means that one is faced with the problem of finding sufficient conditions to allow an isomorphic representation as a subspace of continuous functions on certain topological space. In spaces of continuous real-valued functions on a topological space 𝑄 , the natural pointwise order stands in favourable relations with the vector space or algebraic operations. Since the latter operations are defined pointwise, the space 𝐶(𝑄) turns out to be a vector lattice (see Chapter 2.2). In this book, functions with compact support are characterized abstractly as elements in the vector lattice 𝐶(𝑄). This leads to special elements in an abstract Archimedean vector lattice, the so-called finite, totally finite and selfmajorizing elements. The collections of those elements are the main subject of investigation. The main thrust is to study the existence of nontrivial finite elements in a given vector lattice and in its subspaces, and to describe the structure and properties of such sets. The book is divided into three natural parts: in Chapters 2–4 we provide, apart from the preliminaries, the basic definitions and the main properties of finite, totally finite and selfmajorizing elements in several ambient vector lattices.

1 Introduction

| 3

Chapters 5 and 6 deal with the space of maximal ideals and the topological characterization of the finite, totally finite and selfmajorizing elements. In Chapters 7–10 we investigate the finite elements in vector lattices of continuous functions and deal with various representations of vector lattices as vector lattices of real continuous functions, where the finite elements are represented as finite functions. If the vector lattice has many maximal order ideals and each order ideal can be embedded into a maximal one, then the space of all maximal ideals equipped with a suitable topology carries much information on the vector lattice; in particular the finite elements can be characterized by means of certain compact subsets. An important role in our investigation play the vector lattices of type (Σ), which constitute a natural class of vector lattices and essentially generalize the class of vector lattices with order units. The space of maximal ideals is also used for a representation of vector lattices with a sufficient number of finite elements as vector lattices of continuous functions, where each finite element is represented as a finite function. For vector lattices of type (Σ), the space of maximal ideals has some additional favorable properties which will be applied to the construction of special representations. The results obtained show that the chosen approach turns out to be quite natural. Having finite elements as a new object for studies, the book basically obeys the following lines: – Continuous functions with compact support (finite functions) on a locally compact Hausdorff space and the motivation for finite elements. The main definitions of finite, totally finite and selfmajorizing elements in arbitrary Archimedean vector lattices. Comparison between finite elements and finite functions. – The study of finite and totally finite elements in sublattices, diverse Archimedean vector lattices and Banach lattices. – Finite elements in vector lattices of operators. – The investigation of the space of maximal ideals. – The characterization of finite elements by means of special subsets in the topological space of maximal ideals. – Finite elements in vector lattices of continuous functions. – Representations of vector lattices, where finite elements are represented as continuous functions with compact support. The enumeration of definitions, theorems, propositions, corollaries, lemmas, remarks and examples is specified by chapter. At the end we provide a condensed list of selected examples and counterexamples from the text to help find their treatment quickly in the book.

2 Ordered vector spaces and vector lattices 2.1 Ordered vector spaces and positive operators In this section we collect the necessary basic facts on ordered vector spaces¹, vector lattices, normed Riesz spaces, Banach lattices and operators on these spaces, which we need to present the subject of the book. For a systematic presentation of the theory we refer to the monographs cited at the beginning of Chapter 1, preferably to [9] and [120]. Let 𝑋 be a vector space over the field of real numbers ℝ, and assume that there is a reflexive, antisymmetric and transitive relation ≤ on 𝑋 which is compatible with the vector space operations in the following sense: (1) if for two vectors 𝑥 and 𝑦 of 𝑋 the relation 𝑥 ≤ 𝑦 holds, then also 𝑥 + 𝑧 ≤ 𝑦 + 𝑧 for each vector 𝑧 ∈ 𝑋; (2) if 𝑥 is a vector of 𝑋 such that 𝑥 ≥ 0, then also 𝛼𝑥 ≥ 0 for each nonnegative real number 𝛼. If this is the case, we will write further on 𝑥 ≤ 𝑦 or 𝑦 ≥ 𝑥 and 𝑥 < 𝑦, if 𝑥 ≤ 𝑦 and 𝑥 ≠ 𝑦. Then the pair (𝑋, ≤), or simply 𝑋, is called an ordered vector space. The vectors 𝑥 ∈ 𝑋 with 𝑥 ≥ 0 are called positive. A cone in a vector space 𝑋 is a subset 𝐾 that satisfies the conditions: (i) 𝑥, 𝑦 ∈ 𝐾 and 𝛼, 𝛽 ≥ 0 imply 𝛼𝑥 + 𝛽𝑦 ∈ 𝐾; (ii) 𝑥, −𝑥 ∈ 𝐾 imply 𝑥 = 0. If 𝐾 satisfies only the condition (i), then it is called a wedge. If (𝑋, ≤) is an ordered vector space, then the subset 𝑋+ = {𝑥 ∈ 𝑋 : 0 ≤ 𝑥} is a cone and, on the other hand, if 𝑋 is a vector space and 𝐾 ⊂ 𝑋 a fixed cone, then by defining 𝑥 ≤ 𝑦 as 𝑦 − 𝑥 ∈ 𝐾 one gets an ordered vector space (where the cone 𝑋+ coincides with 𝐾). A cone 𝐾 in 𝑋 is said to be reproducing or generating (for 𝑋) if 𝑋 = 𝐾 − 𝐾. For a nonempty subset 𝐴 in an ordered vector space 𝑋, an element 𝑢 ∈ 𝑋 is an upper bound of 𝐴 if 𝑥 ≤ 𝑢 holds for any 𝑥 ∈ 𝐴. In this case the set 𝐴 is called majorized by 𝑢. Analogously, a lower bound 𝑤 for 𝐴 is defined as 𝑤 ≤ 𝑥 for any 𝑥 ∈ 𝐴 and 𝐴 is called minorized by 𝑤. An order bounded subset 𝐴 is a set which has both an upper and a lower bound. For two elements 𝑎 ≤ 𝑏, the set [𝑎, 𝑏] defined as [𝑎, 𝑏] = {𝑥 ∈ 𝑋 : 𝑎 ≤ 𝑥 ≤ 𝑏} is an order interval. Then a subset 𝐴 of 𝑋 is order bounded if and only if it is included in an order interval.

1 In this book only real vector spaces are considered. By ℕ, ℚ, and ℝ we denote the natural, rational, and real numbers, respectively. For fixed 𝑚 ∈ ℕ and 𝜆 ∈ ℝ there will be used the notations ℕ≥𝑚 := {𝑛 ∈ ℕ : 𝑛 ≥ 𝑚}, ℝ≥𝜆 := {𝛼 ∈ ℝ : 𝛼 ≥ 𝜆}, ℝ>𝜆 := {𝛼 ∈ ℝ : 𝛼 > 𝜆} and ℝ+ for ℝ≥0 . We sometimes write := instead of “defined by”.

2.1 Ordered vector spaces and positive operators |

5

An ordered vector space 𝑋 is said to satisfy the Riesz decomposition property if for any two elements 𝑦, 𝑧 ∈ 𝑋+ [0, 𝑦] + [0, 𝑧] = [0, 𝑦 + 𝑧]. Among all upper bounds of set 𝐸 ⊂ 𝑋 (if there are any at all), there might be a the smallest, i. e., an element 𝑧 ∈ 𝑋 with the two properties 𝑥 ≤ 𝑧 for each 𝑥 ∈ 𝐸, and if 𝑢 ∈ 𝑋 is such that 𝑥 ≤ 𝑢 for all 𝑥 ∈ 𝐸 then 𝑧 ≤ 𝑢. If such 𝑧 exists then it is called the supremum of 𝐸 and is denoted by sup 𝐸. The infimum of a set 𝐸 ≠ 0 is defined in a similar way and is denoted by inf 𝐸. A “natural” (and compatible with the structure of a vector space) order relation is available in many vector spaces. For example, in the vector spaces – ℓ∞ of all real bounded sequences² 𝑥 = (𝑥𝑖)𝑖∈ℕ , where a sequence 𝑥 = (𝑥𝑖 )𝑖∈ℕ 󵄨 󵄨 belongs to ℓ∞ , if and only if there is a real 𝑀 > 0 with the property 󵄨󵄨󵄨𝑥𝑖 󵄨󵄨󵄨 < 𝑀 for all 𝑖 ∈ ℕ; 󵄨󵄨 󵄨󵄨𝑝 – ℓ𝑝 , 1 ≤ 𝑝 ∈ ℕ, of all real sequences 𝑥 = (𝑥𝑖)𝑖∈ℕ , such that ∑∞ 𝑖=1 󵄨󵄨𝑥𝑖 󵄨󵄨 < ∞; – c of all real converging sequences; and in – c0 of all real sequences converging to 0, where the order is the coordinatewise order defined by 𝑥 ≤ 𝑦 ⇔ 𝑥𝑖 ≤ 𝑦𝑖 , ∀𝑖 ∈ ℕ. Very often we will make use of the following ordered vector spaces consisting of functions or classes of functions: – 𝐶(𝑄) of all real-valued continuous functions on a topological space 𝑄, where the order is the pointwise order defined by 𝑥 ≤ 𝑦 ⇔ 𝑥(𝑡) ≤ 𝑦(𝑡) for all 𝑡 ∈ 𝑄; 󵄨 󵄨𝑝 – 𝐿 𝑝 (Ω, Σ, 𝜇) of all classes of 𝜇-equivalent functions such that ∫Ω 󵄨󵄨󵄨𝑓󵄨󵄨󵄨 d𝜇 < ∞, where 1 ≤ 𝑝 ∈ ℕ and (Ω, Σ, 𝜇) is a measure space. The order is defined by 𝑓 ≤ 𝑔 if and only if 𝑓(𝑡) ≤ 𝑔(𝑡) for 𝜇-almost every 𝑡 ∈ Ω; – 𝐿 ∞ (Ω, Σ, 𝜇) of all essentially 𝜇-bounded (classes of 𝜇-equivalent) functions, i. e., 󵄨 󵄨 𝑓 belongs to 𝐿 ∞ (Ω, Σ, 𝜇) if there is a constant 𝑀 such that 󵄨󵄨󵄨𝑓(𝑡)󵄨󵄨󵄨 ≤ 𝑀 for 𝜇-almost every 𝑡 ∈ Ω. The order is defined as in 𝐿 𝑝 (Ω, Σ, 𝜇). For 𝐿 𝑝 (Ω, Σ, 𝜇) and 𝐿 ∞ (Ω, Σ, 𝜇) we also use the notation 𝐿 𝑝 (𝜇) and 𝐿 ∞ (𝜇) respectively. If (𝑎, 𝑏) ⊂ ℝ and 𝜇 is the Lebesgue measure on ℝ, then we write 𝐿 𝑝 (𝑎, 𝑏). A net (𝑥𝛼 )𝛼∈𝐴 in an ordered vector space 𝑋 is said to be decreasing (written as 𝑥𝛼 ↓) if 𝛼 ≤ 𝛽 implies 𝑥𝛼 ≥ 𝑥𝛽 . We write 𝑥𝛼 ↓ 𝑥 if 𝑥𝛼 ↓ and inf 𝛼 𝑥𝛼 = 𝑥 hold. The meaning of 𝑥𝛼 ↑ (increasing net) and 𝑥𝛼 ↑ 𝑥 is analogous. For two ordered vector spaces (𝑋, 𝑋+ ) and (𝑌, 𝑌+ ) denote by L (𝑋, 𝑌) the set of all linear operators from 𝑋 into 𝑌. In L (𝐸, 𝐹) we consider the usual operations of addition and scalar multiplication amongst linear operators.

2 This vector lattice is also denoted by m.

6 | 2 Ordered vector spaces and vector lattices Definition 2.1. An operator 𝑇 ∈ L (𝑋, 𝑌) is called – positive (denoted by 𝑇 ≥ 0) if 𝑇𝑥 ∈ 𝑌+ for all 𝑥 ∈ 𝑋+ ; – regular if 𝑇 = 𝑇1 − 𝑇2 for some positive operators 𝑇1 , 𝑇2 ; – bipositive or order homomorphism if 𝑥 ∈ 𝑋+ if and only if 𝑇𝑥 ∈ 𝑌+ ; – order isomorphism if 𝑇 is surjective and bipositive; – order bounded if 𝑇 maps any order interval of 𝑋 into an order interval of 𝑌; – order continuous if 0 ≤ 𝑥𝛼 ↑ 𝑥 in 𝑋 implies 𝑇𝑥𝛼 ↑ 𝑇𝑥 in 𝑌. The set of all positive operators in L (𝑋, 𝑌) is denoted by L+ (𝑋, 𝑌), and by L+ (𝑋) if 𝑋 = 𝑌. In general L+ (𝑋, 𝑌) is a wedge. If the cone 𝑋+ is generating in 𝑋 then L+ (𝑋, 𝑌) is a cone in L (𝑋, 𝑌), and so the pair (L (𝑋, 𝑌), L+ (𝑋, 𝑌)) is an ordered vector space. For 𝑆, 𝑇 ∈ L (𝑋, 𝑌) we write 𝑆 ≤ 𝑇 if 𝑇 − 𝑆 is positive. The sets of all regular, order bounded, and order continuous operators 𝑇 : 𝑋 → 𝑌 are denoted by L 𝑟 (𝑋, 𝑌), L 𝑏 (𝑋, 𝑌), and L 𝑛 (𝑋, 𝑌) respectively, and their intersections with L+ (𝑋, 𝑌) by L+𝑟 (𝑋, 𝑌), L+𝑏 (𝑋, 𝑌), L+𝑛(𝑋, 𝑌) respectively. Each positive operator is regular and each regular operator is order bounded. Therefore, in general, L+(𝑋, 𝑌) ⊂ L 𝑟 (𝑋, 𝑌) ⊆ L 𝑏 (𝑋, 𝑌). In case of 𝑌 = 𝑋, we write L 𝛼 (𝑋) instead of L 𝛼 (𝑋, 𝑋) for 𝛼 = 𝑟, 𝑏, 𝑛.

2.2 Vector lattices If the order in an ordered vector space 𝐸 has the property that any set {𝑥, 𝑦} consisting of two elements 𝑥, 𝑦 ∈ 𝐸 possesses both its supremum and its infimum then 𝐸 is called a vector lattice or a Riesz space. For the elements sup{𝑥, 𝑦} and inf{𝑥, 𝑦}, the usual notations are 𝑥 ∨ 𝑦 and 𝑥 ∧ 𝑦, respectively. Let 𝐸 be vector lattice. Since now for each element 𝑥 ∈ 𝐸, the elements 𝑥+ := 𝑥 ∨ 0 and 𝑥− := (−𝑥) ∨ 0 (the positive and negative parts of 𝑥, respectively) exist, 𝑥 has the representation 𝑥 = 𝑥+ − 𝑥− . For each element 𝑥 the element |𝑥| := 𝑥+ ∨ 𝑥− is called the modulus or absolute value of 𝑥. If each two-point set of 𝐸 has its supremum then the supremum also exists for any finite set of vectors. It will be denoted by 𝑥1 ∨ . . . ∨ 𝑥𝑛 or ⋁𝑛𝑖=1 𝑥𝑖 . Analogously, 𝑥1 ∧, . . . , 𝑥𝑛 or ⋀𝑛𝑖=1 𝑥𝑖 denote the infimum of the set {𝑥1 , . . . , 𝑥𝑛 }. If 𝐴 is a subset of a vector lattice for which sup(𝐴) exists, then the elements sup(−𝐴), inf(𝐴) also exist and inf(𝐴) = − sup(−𝐴). Definition 2.2 (Archimedean vector lattice). A vector lattice 𝐸 is said to be Archimedean, if 𝑥, 𝑦 ∈ 𝐸 and 𝑛𝑥 ≤ 𝑦 for all 𝑛 ∈ ℕ imply 𝑥 ≤ 0. The Archimedean property of a vector lattice is useful because it is equivalent to the following: for every 0 < 𝑦 ∈ 𝐸 there holds 𝛼𝑛 𝑦 ↓ 0, whenever 𝛼𝑛 is a sequence of real numbers with 𝛼𝑛 ↓ 0 (see e. g., [84, § 22]).

2.2 Vector lattices

| 7

Definition 2.3 (Dedekind completeness). A vector lattice 𝐸 is said to be Dedekind complete, if each nonempty set 𝐴 ⊆ 𝐸 which is bounded from above possesses its supremum in 𝐸. A vector lattice 𝐸 is said to be 𝜎-Dedekind complete, if each nonempty countable upper bounded subset 𝐴 ⊆ 𝐸 possesses its supremum in 𝐸. For a vector lattice 𝐸, the following conditions are equivalent (see [111]): (i) 𝐸 is Dedekind complete; (ii) every net (𝑥𝛼 )𝛼∈𝐴 with 0 ≤ 𝑥𝛼 ↑ ≤ 𝑥 possesses a supremum; (iii) every net (𝑥𝑎 ) with 𝑥𝛼 ↓ ≥ 0 possesses an infimum. A Dedekind complete vector lattice is 𝜎-Dedekind complete and any 𝜎-Dedekind complete vector lattice is always Archimedean, i. e. the following implications hold: Dedekind complete 󳨐⇒ 𝜎-Dedekind complete 󳨐⇒ Archimedean. Every Archimedean vector lattice 𝐸 possesses a unique Dedekind completion, i. e., there exist a uniquely determined up to a lattice isomorphism³ Dedekind complete vector lattice 𝐸𝛿 and a lattice isomorphism 𝑗 : 𝐸 → 𝐸𝛿 , such that 𝐸 is vector lattice isomorphic to a vector sublattice of 𝐸𝛿 , and for each 𝑧 ∈ 𝐸𝛿 one has (2.1)

𝑧 = sup{𝑥 ∈ 𝐸 : 𝑥 ≤ 𝑧} = inf{𝑦 ∈ 𝐸 : 𝑧 ≤ 𝑦}.

For simplicity we identify E with the subset 𝑗(𝐸) of 𝐸𝛿 and say 𝐸 is embedded in 𝐸𝛿 . The embedding 𝑗 of 𝐸 in 𝐸𝛿 preserves the suprema and infima, i. e., whenever 𝑢 = sup{𝑥 : 𝑥 ∈ 𝐴} in 𝐸 exists for some subset 𝐴 ⊂ 𝐸, then 𝑗(𝑢) = sup{𝑗(𝑥) : 𝑥 ∈ 𝐴} holds in 𝐸𝛿 , and analogously for infima. For details see [84, 120]. Further on all vector lattices will be assumed to be Archimedean. Recall some definitions, notations, and elementary facts in an Archimedean vector lattice (𝐸, 𝐸+ ) which will be used further on. In most cases we refer to [9, 95, 120]. – A net (𝑥𝛼 )𝛼∈𝐴 in 𝐸 is said to be order convergent or (𝑜)-convergent to 𝑥 whenever a (decreasing) net (𝑦𝛼 )𝛼∈𝐴 exists (with the same index set), such that 𝑦𝛼 ↓ 0 and (𝑜)

–

|𝑥𝛼 −𝑥| ≤ 𝑦𝛼 for all 𝛼. This is written: 𝑥𝛼 󳨀→ 𝑥 and 𝑥 is called the (𝑜)-limit of (𝑥𝛼 )𝛼∈𝐴 . A net (𝑥𝛼 )𝛼∈𝐴 in 𝐸 is said to be uniformly convergent⁴ or (𝑟)-convergent to 𝑥 if there exist an element 𝑦 ∈ 𝐸+ (a regulator of convergence) and a net (𝜀𝛼 )𝛼∈𝐴 of positive (𝑟)

numbers, such that 𝜀𝛼 → 0 and |𝑥𝛼 − 𝑥| ≤ 𝜀𝛼 𝑦. This is written: 𝑥𝛼 󳨀→ 𝑥 and 𝑥 is called the (𝑟)-limit of (𝑥𝛼 )𝛼∈𝐴 . Analogously, the uniform convergence of a sequence (𝑟)

(𝑜)

is defined. In an Archimedean vector lattice, 𝑥𝛼 󳨀→ 𝑥 implies 𝑥𝛼 󳨀→ 𝑥, and the (𝑟)limit is uniquely defined (see [120]).

3 For the notion of a lattice isomorphism see at the end of this subparagraph. 4 Or convergent with a regulator.

8 | 2 Ordered vector spaces and vector lattices –

–

A uniformly Cauchy sequence or (𝑟)-Cauchy sequence in 𝐸 is a sequence (𝑥𝑛)𝑛∈ℕ for which a regulator 𝑦 ∈ 𝐸+ exists, such that for each 𝜀 > 0 there is a number 𝑛𝜀 with |𝑥𝑛 − 𝑥𝑚 | ≤ 𝜀𝑦 for all 𝑛, 𝑚 ∈ ℕ with 𝑛, 𝑚 ≥ 𝑛𝜀. An Archimedean vector lattice 𝐸 is called uniformly complete or (𝑟)-complete if each uniformly Cauchy sequence in 𝐸 is uniformly convergent. Two elements 𝑥, 𝑦 ∈ 𝐸 are called disjoint, written as 𝑥 ⊥ 𝑦, if |𝑥| ∧ |𝑦| = 0. For any nonempty subset 𝐴 ⊂ 𝐸 define the set: 𝐴⊥ = {𝑥 ∈ 𝐸 : 𝑥 ⊥ 𝑦 for any 𝑦 ∈ 𝐴}.

– – –

A subset 𝐴 ⊂ 𝐸 is called complete if 𝑥 ⊥ 𝐴 implies 𝑥 = 0. This is also written as 𝐴⊥ = {0}. A subset 𝐴 ⊂ 𝐸 is called solid (sometimes also called normal), if 𝑥 ∈ 𝐸, 𝑎 ∈ 𝐴 and |𝑥| ≤ |𝑎| implies 𝑥 ∈ 𝐴. A linear subspace 𝐼 of a vector lattice 𝐸 is said to be an 𝑙-ideal, or simply an ideal, if 𝐼 is solid. Clearly, {0} and 𝐸 are always ideals, the so-called trivial ideals. If 𝐴 is a nonempty subset of 𝐸, then the smallest ideal that contains 𝐴 is denoted by 𝐼𝐴 and is called the ideal generated by 𝐴. This order ideal is (see [9]): 𝑛

󵄨 󵄨 𝐼𝐴 = {𝑥 ∈ 𝐸 : ∃𝑎1 , . . . , 𝑎𝑛 ∈ 𝐴 and 𝜆 1 , . . . , 𝜆 𝑛 ∈ ℝ+ such that |𝑥| ≤ ∑ 𝜆 𝑖 󵄨󵄨󵄨𝑎𝑖 󵄨󵄨󵄨 }. 𝑖=1

If 𝐴 consists of one element 𝑥 ∈ 𝐸, the ideal 𝐼𝑥 := 𝐼{𝑥} = {𝑦 ∈ 𝐸 : ∃𝜆 ≥ 0, such that |𝑦| ≤ 𝜆|𝑥|} – –

–

–

is called the principal ideal (generated by the element 𝑥). A set 𝐵 ⊂ 𝐸 is called a band if it is an order closed ideal, that is the limit (in 𝐸) of any order convergent net of the ideal 𝐵 belongs to 𝐵. The set 𝐴⊥⊥ is known as the band generated by 𝐴; it is the smallest band that contains 𝐴. If 𝐴 consists of one single element 𝑥, the band generated by {𝑥} is denoted by {𝑥}⊥⊥ and called the principal band (generated by the element 𝑥). A band 𝐵 in 𝐸 is said to be a projection band if 𝐸 = 𝐵 ⊕ 𝐵⊥ . In this case any element 𝑥 ∈ 𝐸 has a unique representation 𝑥 = 𝑥1 + 𝑥2 , where 𝑥1 ∈ 𝐵 and 𝑥2 ∈ 𝐵⊥ . The map 𝑃𝐵 : 𝐸 → 𝐸 defined by 𝑃𝐵 (𝑥) = 𝑥1 for any 𝑥 ∈ 𝐸 = 𝐵 ⊕ 𝐵⊥ is a positive projection. In a Dedekind complete vector lattice any band is a projection band. If {𝑢}⊥⊥ is a projection band, then 𝑃{𝑢} is denoted by 𝑃𝑢 and is called the band projection. In this case for each element 𝑥 ≥ 0 the element sup{𝑥 ∧ 𝑛|𝑢|} exists, and 𝑃𝑢 (𝑥) (for 𝑥 ≥ 0) is calculated by the formula: 𝑃𝑢 (𝑥) = sup{𝑥 ∧ 𝑛|𝑢|} .

–

(2.2)

A vector lattice 𝐸 is said to have the principal projection property (𝑝𝑝𝑝), if {𝑥}⊥⊥ is a projection band for each 𝑥 ∈ 𝐸. Any 𝜎-Dedekind complete vector lattice has the (𝑝𝑝𝑝), and (𝑝𝑝𝑝) implies that 𝐸 is Archimedean (see [144, Theorems 11.9 and 12.3]), i.e. the following implication holds: 𝜎-Dedekind complete 󳨐⇒ (𝑝𝑝𝑝) 󳨐⇒ Archimedean.

2.2 Vector lattices

–

– – –

– –

|

9

An element 𝑢 ∈ 𝐸+ , 𝑢 ≠ 0 is an order unit⁵, if for each 𝑥 ∈ 𝐸 there is a 𝜆 ∈ ℝ>0 with −𝜆𝑢 ≤ 𝑥 ≤ 𝜆𝑢 (or equivalently, |𝑥| ≤ 𝜆𝑢). A vector lattice with an order unit is called a vector lattice of bounded elements. Let 𝐸 be an Archimedean vector lattice, and 0 < 𝑢 a fixed positive element in 𝐸. Then the principal ideal 𝐼𝑢 is a vector lattice of bounded elements in 𝐸. An element 𝑒 ∈ 𝐸+ , 𝑒 ≠ 0 is a weak order unit, if 𝑥 ∈ 𝐸 and 𝑥 ⊥ 𝑒 imply 𝑥 = 0, i. e., {𝑒}⊥⊥ = 𝐸. An element 𝑎 ∈ 𝐸+ , 𝑎 ≠ 0 is called an atom of 𝐸 whenever 0 < 𝑏, 𝑐 ≤ 𝑎, and 𝑏 ∧𝑐 = 0 implies that either 𝑏 = 0 or 𝑐 = 0. An element 𝑎 ∈ 𝐸+ , 𝑎 ≠ 0 is called a discrete element whenever 0 ≤ 𝑏 ≤ 𝑎 implies 𝑏 = 𝜆𝑎 for some 𝜆 ∈ ℝ+ . In an Archimedean Riesz space a positive element is an atom if and only if it is a discrete element. The principal band {𝑎}⊥⊥ (in an Archimedean Riesz space) generated by an atom 𝑎 consists of all real multiples of 𝑎 and is a projection band (see [2, § 2.3], [84], [120, § III.13], [141]). A vector lattice is said to be atomic, if for each 𝑥 > 0 there is an atom 𝑎, such that 0 < 𝑎 ≤ 𝑥; i.e., the set of all atoms is a complete subset.. The sequence spaces c0 , c and ℓ𝑝 for 1 ≤ 𝑝 ≤ ∞ are atomic vector (even Banach) lattices. Each vector lattice satisfies the Riesz decomposition property (see [95, Theorem 1.1.1]). A vector lattice 𝐸 not possessing any order unit is called of type (Σ) if 𝐸 contains a sequence of elements (𝑒𝑛)∞ 𝑛=1 with the following property: (Σ󸀠 )

𝑒1 ≤ 𝑒2 ≤ ⋅ ⋅ ⋅ ≤ 𝑒𝑛 ≤ ⋅ ⋅ ⋅ , { for any 𝑥 ∈ 𝐸 there exist 𝑛 ∈ ℕ and 𝐶 > 0 such that |𝑥| ≤ 𝐶𝑒𝑛 .

The set L 𝑟 (𝐸, 𝐹) is the linear span of all positive operators from 𝐸 into 𝐹. If 𝐹 is Dedekind complete, then L 𝑟 (𝐸, 𝐹) is a vector lattice. The subsequent theorem is of great importance within the whole theory of vector lattices. It is crucial and decisive for our investigation in Chapter 4. Theorem 2.4 (F. Riesz, L. V. Kantorovich). Let 𝐸 and 𝐹 be vector lattices with 𝐹 Dedekind complete. Then the ordered vector space L 𝑟 (𝐸, 𝐹) is a Dedekind complete vector lattice satisfying L 𝑟 (𝐸, 𝐹) = L 𝑏 (𝐸, 𝐹). The lattice operations in L 𝑟 (𝐸, 𝐹) are given by the formulas (1) 𝑇+ (𝑥) = sup{𝑇𝑦 : 0 ≤ 𝑦 ≤ 𝑥}, (2) 𝑇− (𝑥) = sup{−𝑇𝑦 : 0 ≤ 𝑦 ≤ 𝑥}, (3) |𝑇| (𝑥) = sup{𝑇𝑦 : − 𝑥 ≤ 𝑦 ≤ 𝑥},

5 Very often strong order unit.

10 | 2 Ordered vector spaces and vector lattices (4) (𝑆 ∨ 𝑇)(𝑥) = sup{𝑆(𝑥1 ) + 𝑇(𝑥2 ) : 𝑥1 , 𝑥2 ∈ 𝐸+ , 𝑥 = 𝑥1 + 𝑥2 }, (5) (𝑆 ∧ 𝑇)(𝑥) = inf{𝑆(𝑥1 ) + 𝑇(𝑥2 ) : 𝑥1 , 𝑥2 ∈ 𝐸+ , 𝑥 = 𝑥1 + 𝑥2 }, for all 𝑇, 𝑆 ∈ L 𝑟 (𝐸, 𝐹) and all 𝑥 ∈ 𝐸+ . The formulas (1)–(5) are usually called the Riesz–Kantorovich formulas, see [2, Theorem 1.16], [9, Theorem 1.13], [120, Theorem VIII.2.1], and [95, § 1.3]. In [3] there is proved the converse of that theorem. Theorem 2.5. For Archimedean vector lattices 𝐸, 𝐹 the following statements are equivalent: (1) 𝐹 is Dedekind complete; (2) the equality L 𝑏 (𝐸, 𝐹) = L 𝑟 (𝐸, 𝐹) holds for every vector lattice 𝐸 and L 𝑟 (𝐸, 𝐹) is a Dedekind complete vector lattice; (3) the equality L 𝑏 (𝐸, 𝐹) = L 𝑟 (𝐸, 𝐹) holds for every vector lattice 𝐸, and L 𝑟 (𝐸, 𝐹) is a vector lattice. In general, the regular operators need not be a vector lattice, e. g., L 𝑟 (𝐿 1 ([0, 1]), c) is not a vector lattice (see [105]). If 𝐹 = ℝ, then the vector space L 𝑏 (𝐸, ℝ) of all order bounded functionals on 𝐸 is called the order dual of 𝐸, and is denoted by 𝐸̃ . Due to the preceding theorem 𝐸̃ is always a Dedekind vector lattice. Let 𝐸, 𝐹 be two vector lattices. An operator 𝑇 ∈ L (𝐸, 𝐹) is called lattice homomorphism or Riesz homomorphism of 𝐸 into 𝐹 if it preserves the lattice operations, i. e., 𝑇(𝑥 ∨ 𝑦) = 𝑇𝑥 ∨ 𝑇𝑦

and 𝑇(𝑥 ∧ 𝑦) = 𝑇𝑥 ∧ 𝑇𝑦

for all 𝑥, 𝑦 ∈ 𝐸.

If this is the case, then one also has |𝑇(𝑥)| = 𝑇(|𝑥|), and 𝑇 ≥ 0, i. e., the operator 𝑇 is positive. Any Riesz homomorphism 𝑇 is (𝑟)-continuous, i. e., continuous with re󵄨 󵄨 spect to the (𝑟)-convergence. This follows immediately from the relations 󵄨󵄨󵄨𝑇𝑥𝛼 − 𝑇𝑥󵄨󵄨󵄨 = 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑇(𝑥𝛼 − 𝑥)󵄨󵄨 = 𝑇(󵄨󵄨𝑥𝛼 − 𝑥󵄨󵄨), and 𝑇(󵄨󵄨𝑥𝛼 − 𝑥󵄨󵄨) ≤ 𝜀𝛼 𝑇𝑦. A lattice homomorphism 𝑇 is a lattice isomorphism or a Riesz isomorphism if 𝑇 is bijective⁶. In this case the operator 𝑇−1 exists, and is a lattice isomorphism of 𝐹 onto 𝐸 as well. Obviously, lattice homomorphisms and lattice isomorphisms are order homomorphisms and order isomorphisms as in Definition 2.1. Two vector lattices 𝐸, 𝐹 are called lattice isomorphic if there exists a lattice isomorphism between 𝐸 and 𝐹. An order continuous lattice homomorphism preserves the exact bounds, i. e., if for a subset 𝐴 ⊂ 𝐸 there exists the supremum in 𝐸 then the supremum exists also for the set 𝑇(𝐴) in 𝐹 and, 𝑇(sup 𝐴) = sup 𝑇(𝐴). The analogous statement holds for the infimum.

6 I. e., a one-to-one map of 𝐸 onto 𝐹.

2.3 Ordered normed spaces

| 11

2.3 Ordered normed spaces In many classical normed spaces a natural partial order exists which, as shown previously, may be introduced by means of a certain cone. This gave rise to dealing with this special class of spaces on the one hand as part of the theory of normed spaces and on the other hand as part of partially ordered vector spaces. The first investigations go back to the 1930s and were related to problems concerning the positivity of operators. After World War Two, the initial impulse for a stormy development of this theory was set off by the famous article by M. G. Krein and M. A. Rutman [69]. Of course, formally ordered normed spaces are equipped with two structures: the structure of a normed (vector) space and the one of an ordered vector space. For a rich theory, but essentially also for applications, the compatibility between both structures is formulated in terms of topological properties of the cone in such spaces. The theory has been systematically developed in several directions by many authors. For the treatment in ordered normed spaces, see e. g., [65, 66, 106, 121, 122], for linear topological spaces, see e. g., [106]. Today, together with the theory of Banach lattices, the theory of ordered normed spaces is one of the pillars of the concept of positivity, not only in functional analysis but also in other branches of mathematics (e. g., integral and measure theory, numerical analysis, dynamical systems, optimization, economy, operator theory and more). For a normed space (𝑋, ‖⋅‖) the set 𝐵𝑋 = {𝑥 ∈ 𝑋 : ‖𝑥‖ ≤ 1} is its closed unit ball. The space of all linear continuous (or linear bounded) operators from the normed space (𝑋, ‖⋅‖𝑋 ) into the normed space (𝑌, ‖⋅‖𝑌 ), equipped with the operator norm, i. e., ‖𝑇‖ = sup{𝑇𝑥 : 𝑥 ∈ 𝐵𝑋 }

for 𝑇 : 𝑋 → 𝑌

is denoted by L(𝑋, 𝑌), and by L(𝑋) in the case of 𝑋 = 𝑌. If 𝑌 = ℝ then 𝑋󸀠 = L(𝑋, ℝ) is the space of all linear continuous functionals on 𝑋, the (Banach) norm dual space of 𝑋. With the usual algebraic operations for operators and functionals these spaces turn out to be normed (vector) spaces. The order in L(𝑋, 𝑌) is introduced by means of L+ (𝑋, 𝑌) = L(𝑋, 𝑌) ∩ L+ (𝑋, 𝑌). Let now (𝑋, ‖⋅‖𝑋 , 𝑋+ ) and (𝑌, ‖⋅‖𝑌 , 𝑌+ ) be ordered normed spaces, which for short, we denote by 𝑋 and 𝑌, respectively. Then it is a natural question to ask what the relation between positive and continuous operators is. At a first glance the answer is surprising, the (nontopological) property of positivity implies the topological property of continuity (of course, under some additional conditions concerning the compatibility of the norm and the order). The general result, which in its full extent was proven by I. A. Bakhtin, M. A. Krasnoselskij, V. Ya. Stetsenko and G. Ya. Lozanovskij, is presented below (see [122, Theorems VI.2.1 and VI.2.2]). Theorem 2.6. If (𝑋, ‖⋅‖𝑋 , 𝑋+ ) is an ordered Banach space such that each positive linear functional on 𝑋 is continuous and (𝑌, ‖⋅‖𝑌 , 𝑌+ ) is an ordered Banach space with a closed cone 𝑌+ , then any linear positive operator 𝑇 : 𝑋 → 𝑌 is continuous.

12 | 2 Ordered vector spaces and vector lattices Proof. For completeness we reproduce the simple proof from [122]. Based on the closed graph theorem it suffices to prove that 𝑥𝑛 → 0 (in 𝑋) and 𝑇𝑥𝑛 → 𝑦0 (in 𝑌) imply 𝑦0 = 0. Assume 𝑦0 ≠ 0. If 𝑦0 ∉ 𝑌+ then due to 𝑌+ being closed, the cone 𝑌+ and the point 𝑦0 can be separated by a closed hyperplane {𝑦 ∈ 𝑌 : 𝜑(𝑦) = 𝑐} not containing 𝑦0 , where 𝜑 ∈ 𝑌󸀠 and 𝑐 is a real number (see e. g.,[44, Corollary 2.1.4]), i. e., we may assume 𝜑(𝑦) ≥ 𝑐 for 𝑦 ∈ 𝑌+ and 𝜑(𝑦0 ) ≠ 𝑐. From 0 ∈ 𝑌+ and 𝜑(0) = 0 we have 𝑐 ≤ 0, so 𝜑(𝑦0 ) < 0. On the other hand, 𝜑(𝑦) = 𝑛1 𝜑(𝑛𝑦) ≥ 𝑛𝑐 󳨀→ 0 for arbitrary 𝑦 ∈ 𝑌+ , 𝑛→∞

which implies 𝜑(𝑦) ≥ 0, and therefore 𝜑 ∈ 𝑌+󸀠 . The functional 𝑓 = 𝜑 ∘ 𝑇 is linear and positive on 𝑋 and so, by condition, continuous. Therefore 𝑓(𝑥𝑛) → 0. However, 𝑓(𝑥𝑛) = 𝜑(𝑇𝑥𝑛) 󳨀→ 𝜑(𝑦0 ) ≠ 0. The contradiction shows that the operator 𝑇 is continuous. The case 𝑦0 ∈ 𝑌+ reduces to the previous one because of −𝑦0 ∉ 𝑌+. Throughout, using int (𝐴), we denote the set of all interior points of a subset 𝐴 in a topological space. The condition for 𝑋 in the previous theorem holds e. g., if int (𝑋+ ) ≠ 0, or if the cone 𝑋+ is closed and reproducing. The norm-completeness of 𝑌 can be removed if the cone 𝑌+ is assumed to be normal in 𝑌 (see [122, VI]). An important consequence of this theorem is the following result. We formulate it here as: Corollary 2.7. Let 𝐸 be an ordered vector space with a generating cone 𝐸+ . If ‖⋅‖1 and ‖⋅‖2 are two given norms on 𝐸 under each of which 𝐸 is a Banach space and the cone 𝐸+ is closed, then the norms are equivalent. Proof. Indeed, the identity operator which maps one space of (𝐸, ‖⋅‖𝑖 ), 𝑖 = 1, 2 onto the other one is positive and in view of the theorem, continuous. Therefore, the relations 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑥𝑛󵄩󵄩𝑖 󳨀→ 0 for 𝑖 = 1, 2 are equivalent. 𝑛→∞

For the case of a normed Riesz space, this result is known as the Nakano–Makarov Theorem. For Banach lattices, both the theorem and the corollary for further use are reformulated as Theorem 2.12 and Corollary 2.13 in Section 2.4.

2.4 Normed Riesz spaces and Banach lattices Ordered normed spaces and ordered Banach space are very often vector lattices. In this case, the relation between the lattice structure and the norm usually satisfies the condition ‖𝑥‖ ≤ ‖𝑦‖ whenever |𝑥| ≤ |𝑦|, (2.3)

2.4 Normed Riesz spaces and Banach lattices

|

13

e. g., in the spaces 𝐶(𝑄), 𝑙𝑝 and 𝐿 𝑝 (𝜇). If the norm ‖ ⋅ ‖ in a vector lattice satisfies the condition (2.3), then it is called a lattice norm or⁷ a Riesz norm. A vector lattice (i. e., a Riesz space) equipped with a Riesz norm is said to be a normed vector lattice or a normed Riesz space. A normed Riesz space which is complete with respect to its norm is called a Banach lattice. The compatibility of the two structures (vector lattice and normed space) on a normed Riesz space (𝐸, ‖⋅‖) postulated by (2.3) has some valuable consequences, e. g., a) the unit ball (and so also any other ball with center at zero) is a solid subset; b) for any 𝑥 ∈ 𝐸 one has ‖𝑥‖ = ‖|𝑥|‖; c) each normed vector lattice is Archimedean; d) the lattice operations in a normed vector lattice are ‖⋅‖-continuous, in particular, ‖𝑥𝑛 − 𝑥‖ → 0 and ‖𝑦𝑛 − 𝑦‖ → 0 imply ‖𝑥𝑛 ∨ 𝑦𝑛 − 𝑥 ∨ 𝑦‖ → 0; e) the cone 𝐸+ = {𝑥 ∈ 𝐸 : 𝑥 ≥ 0} is norm closed. The lattice operations in a Banach lattice 𝐸 are said to be weakly sequentially contin𝑤 󵄨 󵄨 𝑤 uous, if 󵄨󵄨󵄨𝑥𝑛󵄨󵄨󵄨 󳨀→ 0 whenever 𝑥𝑛 󳨀→ 0, where “𝑤” stands for the weak topology 𝜎(𝐸, 𝐸󸀠 ) in 𝐸. The norm in a normed lattice is called order continuous if the implication holds that 𝑥𝛼 ↓ 0 󳨐⇒ ‖𝑥𝛼 ‖ → 0. Many characterizations of the order continuity of a norm in Banach lattices are provided in [139, Chapter I]. In particular, a Banach lattice with order continuous norm is always Dedekind complete. In the proof of Theorem 3.18 we use the following result (see [139, Theorem 6.1], [133, Theorem 5]). Theorem 2.8. A Banach lattice 𝐸 is atomic and the norm on 𝐸 is order continuous if and only if the order intervals in 𝐸 are norm compact. If in an atomic Banach lattice the norm is order continuous, then the lattice operations are weakly sequentially continuous (see [95, Proposition 2.5.33]) Clearly, c0 , c and ℓ𝑝 (1 ≤ 𝑝 ≤ ∞) are atomic Banach lattices, however, only in c0 and ℓ𝑝 are the norms order continuous. In the Soviet literature up to the late 1980s, the order continuity of the norm was named condition (A), see e. g., [59, § X.4], [120, § VII.6], [122, § I.5], and also introductory remarks in [81, 82]. If a locally solid topology 𝜏 on a vector lattice satisfies the implication 𝜏

𝑥𝛼 ↓ 0 󳨐⇒ 𝑥𝛼 󳨀→ 0, then the topology 𝜏 is called a Lebesgue topology.

7 Sometimes also known as monotone norm.

14 | 2 Ordered vector spaces and vector lattices Definition 2.9. A Banach lattice 𝐸 is said to be an – 𝐴𝑀-space if the norm satisfies the condition ‖𝑥 ∨ 𝑦‖ = max{‖𝑥‖, ‖𝑦‖} –

(2.4)

𝐴𝐿-space if the norm satisfies the condition ‖𝑥 + 𝑦‖ = ‖𝑥‖ + ‖𝑦‖

–

𝑥, 𝑦 ∈ 𝐸+ ,

𝑥, 𝑦 ∈ 𝐸+ ,

abstract 𝐿 𝑝 -space⁸ (1 ≤ 𝑝 < ∞), if the norm satisfies the condition ‖𝑥 + 𝑦‖𝑝 = ‖𝑥‖𝑝 + ‖𝑦‖𝑝

for any disjoint 𝑥, 𝑦 ∈ 𝐸.

For 1 ≤ 𝑝 < ∞, the Banach (function) lattices 𝐿 𝑝 (𝜇) and their abstractions, the abstract 𝐿 𝑝 -spaces, have order continuous norms (see [2, Corollary 3.7], [9, Section 12]), but the norm in 𝐶(𝐾) fails to be order continuous except in the case that 𝐾 is finite (see e. g., [95, § 2.4]). Each 𝐴𝑀-space fails to possess an order continuous norm as well (see [95]). In any 𝐴𝑀-space the lattice operations are weakly sequentially continuous (see [95, Proposition 2.1.11], and [9, Theorem 12.30]). Let 𝐸 be an Archimedean vector lattice and 0 < 𝑢. Then in the ideal 𝐼𝑢 = {𝑥 ∈ 𝐸 : |𝑥| ≤ 𝜆𝑢} by means of the formula ‖𝑥‖𝑢 = inf{𝜆 > 0 : |𝑥| ≤ 𝜆𝑢},

(2.5)

a norm is defined which is called u-norm or order unit norm. Then (𝐼𝑢 , ‖⋅‖𝑢 ) is a normed Riesz space with [−𝑢, 𝑢] as the closed unit ball, where the norm ‖⋅‖ satisfies the equation (2.4). If 𝐸 is a Banach lattice then the ideal 𝐼𝑢 equipped with the 𝑢-norm is even an 𝐴𝑀-space (see [95, Proposition 1.2.13]). There is an important duality between 𝐴𝐿-spaces and 𝐴𝑀-spaces. The following results will be used later in Chapter 4. There proofs can be found e. g., in [9, Theorem 12.22], [2, Theorem 3.3], [95, Proposition 1.4.7], and [8, Corollary 8.36]. Proposition 2.10. A Banach lattice 𝐸 is an 𝐴𝐿-space (an 𝐴𝑀-space) if and only if its norm dual 𝐸󸀠 is an 𝐴𝑀-space (an 𝐴𝐿-space). Moreover, if 𝐸 is an 𝐴𝐿-space, then 𝐸󸀠 is a Dedekind complete 𝐴𝑀-space with an order unit 𝑢󸀠 , such that 𝑢󸀠 (𝑥) = ‖𝑥+ ‖ − ‖𝑥− ‖ for each 𝑥 ∈ 𝐸. Proposition 2.11. An 𝐴𝐿-space is lattice isomorphic to an 𝐴𝑀-space if and only if it is finite-dimensional. For Banach lattices which constitute, besides the Archimedean vector lattices, the main class of spaces which we will deal with in this book, we obtain from Theorem 2.6 the following interesting and very important results on continuity of positive operators in a special case (see [9, Theorem 12.2], and [95, Proposition 1.3.5]).

8 For 𝑝 = 1 these are exactly the 𝐴𝐿-spaces.

2.4 Normed Riesz spaces and Banach lattices

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15

Theorem 2.12. If 𝐸 is a Banach lattice and 𝐹 a normed vector lattice, then each positive operator is continuous. It follows immediately that each regular operator is continuous as well, and so L 𝑟 (𝐸, 𝐹) ⊂ L(𝐸, 𝐹). The next corollary is an adapted reformulation of Corollary 2.7, and was independently obtained by C. Goffman, B. M. Makarov and H. Nakano (see [100, Theorem 30.28], [51, 88]). It will be used in Section 4.4. Corollary 2.13. All lattice norms that make a vector lattice into a Banach lattice are equivalent. If 𝐸 is a Banach lattice with an order unit 𝑢, then 𝐸 = 𝐼𝑢 . By the previous corollary the 𝑢-norm is equivalent to the original norm, and (𝐸, ‖⋅‖𝑢 ) becomes an 𝐴𝑀-space, with [−𝑢, 𝑢] as the closed unit ball. Often such renorming of Banach lattices or 𝐴𝑀-spaces with order units is very useful. For a normed Riesz space 𝐸, the norm dual 𝐸󸀠 is an ideal in the order dual 𝐸̃. So each 𝑓 ∈ 𝐸󸀠 is order bounded and the next corollary is clear (see [95, Proposition 1.3.7], [77, § 1.5.2]). Corollary 2.14. The norm dual 𝐸󸀠 and, consequently, the second dual 𝐸󸀠󸀠 and any higher dual of any normed Riesz space 𝐸 is always a Dedekind complete Banach lattice. If 𝐸 is a Banach lattice, then 𝐸̃= 𝐸󸀠 . Conditions for Banach lattices 𝐸 and 𝐹 to have the property L 𝑟 (𝐸, 𝐹) = L(𝐸, 𝐹) with ‖𝑇‖𝑟 = ‖𝑇‖, or for L(𝐸, 𝐹) to be a vector lattice are provided e. g., in [95, Theorem 1.5.11], and [138]. A linear operator 𝑇 on an Archimedean vector lattice 𝐸 is called band preserving if 𝑇(𝐵) ⊆ 𝐵 for each band 𝐵 in 𝐸. The last property is equivalent to the requirement 𝑥 ⊥ 𝑦 ⇒ 𝑇𝑥 ⊥ 𝑦 (see [9, Theorem 8.2]). A band-preserving operator which is orderbounded is called an orthomorphism. The set of all orthomorphisms on the vector lattice 𝐸 is denoted by Orth(𝐸). There are very nice relations between an Archimedean vector lattice 𝐸 and its collection Orth(𝐸). First of all, endow Orth(𝐸) with the pointwise algebraic and lattice operations and with the composition as an associative multiplication. We record some facts concerning the relations between a vector lattice 𝐸 and Orth(𝐸). (1) If 𝐸 is an arbitrary Archimedean vector lattice, then Orth(𝐸) is an Archimedean 𝑓-algebra⁹, where the identity operator is a weak order unit in Orth(𝐸) (see [95, Theorem 3.1.10]). (2) If 𝐸 is Dedekind complete then Orth(𝐸) coincides with the band generated by the identity operator 𝐼 in L 𝑏 (𝐸) (see [9, Theorem 8.11]). 9 For the definition see page 50.

16 | 2 Ordered vector spaces and vector lattices (3) If A is an 𝑓-algebra with a multiplicative unit 𝑒, then A is algebraic and lattice isomorphic to Orth(A), where 𝑒 and 𝐼 correspond to each other (see [95, Theorem 3.1.13]). (4) If 𝐸 is a Banach lattice, then (under the regular norm¹⁰) Orth(𝐸) is an 𝐴𝑀-space with order unit 𝐼 (see [9, Theorem 15.5]).

2.5 Representation of Banach lattices In this section we provide important results on representation of (abstract) normed vector lattices and Banach lattices by means of continuous real-valued functions on some Hausdorff space 𝑄, i. e., each function has a finite value at each point of 𝑄, or by means of integrable functions on some measure space. Under the existence of a (strong) order unit in a vector lattice, the first and very important representation results were already proved in the early 1940s. The next famous theorem is related to the mathematicians S. Kakutani, M. G. Krein, and S. G. Krein, although some contributions go back also to H. Nakano and K. Yosida. It shows that 𝐴𝑀-spaces with unit, in essence, are spaces of type 𝐶(𝑄) for compact 𝑄 (see [2, Theorem 3.6]). Later, in Sections 5.1 and 7.1, we deal with representations of general Archimedean vector lattices, which are not required to be normed Riesz spaces. Theorem 2.15 (S. Kakutani, H. F. Bohnenblust, M. G. Krein, S. G. Krein). A Banach lattice 𝐸 is an 𝐴𝑀-space with order unit 𝑢 if and only if 𝐸 is lattice isometric to some space 𝐶(𝑄) for a unique (up to homeomorphism) compact Hausdorff space 𝑄, where the unit 𝑢 can be identified with the constant function 1 on 𝑄. In the proof of this theorem the compact Hausdorff space on which the representation is constructed is nothing other than the weak∗ -compact subset of all extreme points of the positive part of the unit sphere in the norm dual (see [9, Theorems 12.27, 12.28], and also [106, Theorem 8.5]). For normed vector lattices with order unit the following version of Theorem 2.15 is appropriate for us (see [120, Theorem VII.5.1]). Theorem 2.16. For each normed vector lattice 𝐸 with order unit 1 a unique (up to homeomorphism) compact Hausdorff space 𝑄 and a vector lattice isomorphic isometry 𝑖 exist, such that 𝑖(𝐸) is a norm-dense vector sublattice of 𝐶(𝑄), where 𝑖(1) can be assumed to be the constant single function on 𝑄. If 𝐸 is complete with respect to the order unit norm ‖𝑥‖1 , or if 𝐸 is a Banach lattice, then 𝑖(𝐸) = 𝐶(𝑄).

10 see (4.2).

2.5 Representation of Banach lattices

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17

If 𝐸 lacks an order unit but is a Banach lattice, one has the following version (see [9, Theorem 12.28]). Theorem 2.17. A Banach lattice 𝐸 is an 𝐴𝑀-space if and only if 𝐸 is vector lattice isomorphic and isometric to some closed vector sublattice of 𝐶(𝑄). If the norm in an 𝐴𝑀-space 𝐸 has the Nakano property¹¹ , i. e., for any bounded above subset 𝐴 ⊂ 𝐸+ one has sup{‖𝑎‖ : 𝑎 ∈ 𝐴} = inf{‖𝑢‖ : 𝑢 is an upper bound of 𝐴}, then a representation of 𝐸 on some locally compact Hausdorff space 𝑄 exists (see [98, 99, 134, 137]). For a locally compact Hausdorff space 𝑄, the space 𝐶0 (𝑄) is defined as the set of all continuous real-valued functions on 𝑄 vanishing at infinity, i. e., for any 𝑥 ∈ 𝐶0 (𝑄), and any 𝜀 > 0 there is a compact subset 𝐾𝑥,𝜀 ⊂ 𝑄, such that |𝑥(𝑡)| < 𝜀 if 𝑡 ∉ 𝐾𝑥,𝜀 . Theorem 2.18 (H. Nakano). For a Banach lattice 𝐸, the following conditions are equivalent: (1) 𝐸 is an 𝐴𝑀-space with a Nakano norm; (2) there is a locally compact Hausdorff space 𝑄, such that 𝐸 is isometrically lattice isomorphic to the space 𝐶0 (𝑄). An interesting survey on representation theorems for Archimedean vector lattices and Banach lattices can be found in [134]. From our point of view, representations of vector lattices on locally compact Hausdorff spaces are of great interest, such that the image of each finite element (this important class of special elements in a vector lattices is the main subject of the investigation in this book and will be introduced in the next chapter) is a continuous function with compact support. This will be covered in Chapter 9, where some special results, in particular the Theorem of I. Kawai 9.17, will be proved. For abstract 𝐿 𝑝 -spaces, a representation by means of measurable 𝑝-integrable functions is expected [9, Theorem 12.26]. Theorem 2.19 (S. Kakutani, H. F. Bohnenblust, H. Nakano). A Banach lattice 𝐸 is an abstract 𝐿 𝑝 -space for some (1 ≤ 𝑝 < ∞) if and only if 𝐸 is vector lattice isometric to a space 𝐿 𝑝 (Ω, Σ, 𝜇) for some measure space (Ω, Σ, 𝜇). In the proof of this theorem, the order continuity of the norm is used to construct the measure space and the corresponding lattice isometry (see [9, Theorem 12.26]). For general Banach lattices with order continuous norm, the following result is due to R. J. Nagel (see [95, Theorem 2.7.8]). Theorem 2.20. Let 𝐸 be a Banach lattice with order continuous norm, and let 𝐸 possess a weak order unit. Then a measure space (Ω, Σ, 𝜇) and an ideal 𝐼 ⊂ 𝐿 1 (Ω, Σ, 𝜇) exist, such that 𝐿 ∞ (Ω, Σ, 𝜇) ⊂ 𝐼 and 𝐸 is lattice isomorphic to 𝐼. 11 In this case ‖⋅‖ is said to be a Nakano norm.

3 Finite, totally finite and selfmajorizing elements in Archimedean vector lattices In this chapter we introduce the concept of finite, totally finite and selfmajorizing elements in an Archimedean vector lattice. This notion has to be considered as an abstract analogue of continuous function with compact support in vector lattices of continuous functions on locally compact Hausdorff spaces. The systematic study of those elements and the ideals generated by them is carried out in various Archimedean vector lattices and Banach lattices, in 𝑓-algebras and in lattices of operators. Some of their general properties in arbitrary vector lattices are provided in Section 3.1 and in Banach lattices in Section 3.2. A finite element in a vector lattice need not be a finite element in a vector sublattice. The relations between them are studied in Section 3.3. In Section 3.4 selfmajorizing elements are dealt with, whereas finite elements in 𝑓-algebras and in product algebras are considered in Section 3.5. The notion of a finite and a totally finite element in Archimedean vector lattices 𝐸 was introduced in 1972 by B. M. Makarov and the author. The first results were published in [89].

3.1 Finite and totally finite elements in vector lattices Let 𝐸(𝑄) be a vector lattice of continuous functions on a locally compact (noncompact) topological Hausdorff space 𝑄, i. e., 𝐸(𝑄) ⊂ 𝐶(𝑄) (for example 𝑄 = ℝ1 and 𝐸(𝑄) = 𝐶(𝑄)). Denote the linear subspace of all continuous functions on 𝑄 with a compact support¹ by K(𝑄). These functions are of special interest. For example, the construction of an integral on 𝑄 starts either with a positive linear functional (called integral) defined² on K(𝑄) or, if K(𝑄) is equipped with the (locally convex) topology of the inductive limit with respect to the natural embeddings 𝐶(𝐾) 󳨅→ K(𝑄), for all compact subsets 𝐾 ⊂ 𝑄, with a positive linear continuous functional of the corresponding topological dual (called Radon measure). Then the task is to extend the integral to a larger collection of functions than K(𝑄) [97]. In the representation theory of Banach lattices by means of continuous functions on a locally compact space 𝑄, one would like K(𝑄) to be isomorphic to some dense ideal in the represented Banach lattice (see [108]). One may now ask for an abstract characterization of continuous functions having a compact support. This is easily done for a positive function 𝜑 as follows: the family of the infima of all multiples of 𝜑 with any positive function 𝑥 ∈ 𝐸(𝑄) should be majorized

1 I. e., functions 𝑥 ∈ 𝐶(𝑄) for which the closure of {𝑡 ∈ 𝑄 : 𝑥(𝑡) ≠ 0} is a compact set in 𝑄. 2 The functional takes on nonnnegative values on the positive functions of K(𝑄).

3.1 Finite and totally finite elements in vector lattices

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19

by one and the same function, of course with a constant depending on 𝑥 which is suggested by the pictures in Figure 3.1. In general, the moduli of the functions have to be used in an appropriate general definition. In the sequel the abstract version of this description leads to the Definition 3.1 of a finite element, already in an arbitrary Archimedean vector lattice 𝐸. Definition 3.1. An element 𝜑 ∈ 𝐸 is called finite if there is an element 𝑧 ∈ 𝐸 satisfying the following condition: for any element 𝑥 ∈ 𝐸 a number 𝑐𝑥 > 0 exists such that the following inequality holds: |𝑥| ∧ 𝑛|𝜑| ≤ 𝑐𝑥 𝑧

for all 𝑛 ∈ ℕ.

(3.1)

The element 𝑧 is called an 𝐸-majorant or, briefly, a majorant of the finite element 𝜑. The inequality (3.1) describes the special interaction of (the modulus of) a finite element with all elements of the vector lattice. The set of all finite elements of a vector lattice 𝐸 is denoted by Φ1 (𝐸). Any continuous function with compact support is an intrinsic example of a finite element in the vector lattice 𝐶(𝑄), where 𝑄 is assumed to be a locally compact (or only noncompact) topological Hausdorff space. For a majorant of such a function 𝜑 any continuous function³ can be taken which has a strongly positive infimum on the support of 𝜑 (see Fig. 3.1). A finite element in the vector lattice 𝐶(𝑄) must be a function with compact support. Indeed, if 𝜑 ∈ Φ1 (𝐶(𝑄)) would not have a compact support, then a sequence (𝑡𝑖)𝑖∈ℕ in 𝑄 with |𝜑|(𝑡𝑖 ) > 0 exists, and such that for any compact subset 𝐾 ⊂ 𝑄 there is an index 𝑖𝐾 with 𝑡𝑖 ∉ 𝐾 for all 𝑖 > 𝑖𝐾 . If 𝑧 is a majorant for 𝜑, then 𝑧(𝑡𝑖) > 0 for all 𝑖 ∈ ℕ. ̃ 𝑖 ) = 𝑖 𝑧(𝑡𝑖) for all 𝑖. From The function 0 < 𝑥̃ ∈ 𝐶(𝑄) exists, such that 𝑥(𝑡 (|𝑥|̃ ∧ 𝑛|𝜑|)(𝑡𝑖 ) ≤ 𝑐𝑥̃ 𝑧(𝑡𝑖)

for any 𝑛 ∈ ℕ

we get a contradiction, since the left term in the last inequality is equal to 𝑖 𝑧(𝑡𝑖), whereas the right one is 𝑐𝑥̃ 𝑧(𝑡𝑖 ). So Φ1 (𝐶(𝑄)) = K(𝑄). Observe that here we were interested in the finite elements which have to tend to all elements in the large vector lattice 𝐶(𝑄), and therefore we have a very restrictive condition – the compactness of their support. If we consider a smaller vector lattice 𝐸(𝑄) ⊊ 𝐶(𝑄), then one can expect that continuous functions with noncompact support also might be finite elements in 𝐸(𝑄), see Example 8.15. For more details see Section 8.1. In general, the following relations are possible: (i) Φ1 (𝐸) = 𝐸,

(ii) {0} ≠ Φ1 (𝐸) ⫋ 𝐸,

(iii) {0} = Φ1 (𝐸) ⫋ 𝐸.

An example for the case (ii) is the above considered vector lattice 𝐶(𝑄) for noncompact 𝑄. For the first two cases we provide some more examples, where the underlying

3 In fact, such a function does exist in 𝐶(𝑄); see Section 8.1.

20 | 3 Finite, totally finite and selfmajorizing elements nϕ

ϕ

ϕ

nϕ

x

x

cx z z

x ∧ nϕ

x ∧ nϕ

ϕ

Fig. 3.1. Finite element 𝜑 with a majorant 𝑧.

vector lattices are even sequence spaces. Let c be the vector lattice of all real converging sequences and c0 the vector lattice of all zero sequences. Further, denote by c00 the vector lattice of all real finite sequences⁴, i. e., all sequences with only a finite number of nonzero components. For more properties of these vector lattices see [8, Chaps. 15.2–4]. (i) If 𝑥, 𝜑 ∈ c, then |𝑥| ≤ 𝑐𝑥1 and |𝜑| ≤ 𝑐𝜑 1 for some reals 𝑐𝑥 , 𝑐𝜑 ≥ 0, where 1 denotes the sequence (1, 1, . . .). Then the inequality |𝑥| ∧ 𝑛|𝜑| ≤ |𝑥| ≤ 𝑐𝑥 1 holds for each 𝑛 ∈ ℕ and shows the finiteness of 𝜑 (with 1 as one of its majorants) in the vector lattice c. Therefore, Φ1 (c) = c. The subsequent Theorem 3.6 follows the same argument, and so one also has the Corollary 3.7 below. Similarly, if 𝑥, 𝜑 ∈ c00 , then |𝑥| ≤ 𝑐𝑥 𝑒(𝑙) and |𝜑| ≤ 𝑐𝜑 𝑒(𝑚) for some reals 𝑐𝑥, 𝑐𝜑 ≥ 0, 𝑘-times

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ where 𝑒(𝑘) denotes the sequence (1, 1, ..., 1, 0, 0, 0, ...). Then |𝑥| ∧ 𝑛|𝜑| = (|𝑥1 | ∧ 𝑛|𝜑1 |, ..., |𝑥𝑘 | ∧ 𝑛|𝜑𝑘 |, 0, ...) ≤ 𝑐𝑥 𝑒(𝑘) ≤ 𝑐𝑥 𝑒(𝑚)

for any 𝑛 ∈ ℕ,

where 𝑘 = min{𝑚, 𝑙}. Again it is clear that the element 𝜑 is finite with 𝑒(𝑚) as one of its majorants and Φ1 (c00) = c00 . (ii) Let be 𝜑 ∈ Φ1 (c0) and 𝑧 be one fixed majorant of 𝜑. Obviously, due to 𝑧 = (𝑧𝑖 )𝑖∈ℕ ∈ c0 + , the sequence 𝑥 = (√𝑧𝑖 )𝑖∈ℕ also belongs to c0. Assume that infinitely many coordinates of 𝜑 are nonzero. For those coordinates in particular, one has (𝑥 ∧ 𝑛𝜑)𝑖 = √𝑧𝑖 ∧ 𝑛𝜑𝑖 = √𝑧𝑖 ≤ 𝑐𝑥 𝑧𝑖 if 𝑛 is sufficiently large. The relation 0 < 𝑐1 ≤ √𝑧𝑖 (for infinitely many coordinates 𝑥 of 𝑖) contradicts 𝑥 ∈ c0 . So we have Φ1 (c0) = c00 (see also (a) after Theorem 3.18).

4 Sometimes this sequence space is also denoted by 𝜑.

3.1 Finite and totally finite elements in vector lattices

| 21

As an example for case (iii), we refer to our later Example 3.5. After Theorem 3.18 (in case (b)) we will see that also Φ1 (𝐿 𝑝 [0, 1]) = {0} for 1 ≤ 𝑝 < ∞. In our examples we showed the finiteness of certain elements directly by applying Formula (3.1). Later we will develop more general methods for detecting finite elements in vector lattices. It is easy to see that in the case of a 𝜎-Dedekind complete vector lattice, an element 𝜑 is finite and has the element 𝑧 as its majorant if and only if Pr𝜑 |𝑥| ≤ 𝜆𝑧 for some 𝜆 > 0, i. e., the band {Pr𝜑 𝑥 : 𝑥 ∈ 𝐸} is a vector lattice of bounded elements. For any finite element 𝜑 and its majorant 𝑧, put 󵄨 󵄨 𝑝(𝑥) = 𝑝𝜑,𝑧(𝑥) = inf{𝜆 > 0 : |𝑥| ∧ 𝐶 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 ≤ 𝜆𝑧, ∀ 𝐶 > 0}.

(3.2)

It is clear that for all 𝐶 > 0 the following inequality holds: 󵄨 󵄨 |𝑥| ∧ 𝐶 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 ≤ 𝑝(𝑥)𝑧. The function 𝑝, which is defined by Formula (3.2) turns out to be a seminorm on 𝐸. We make use of them only in Section 9.2. The triangle inequality is seen from 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨𝑥 + 𝑦󵄨󵄨󵄨 ∧ 𝐶 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 ≤ (|𝑥| + 󵄨󵄨󵄨𝑦󵄨󵄨󵄨) ∧ 𝐶 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 ≤ |𝑥| ∧ 𝐶 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ∧ 𝐶 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 ≤ (𝑝(𝑥) + 𝑝(𝑦))𝑧. The proof of homogeneity is also elementary: for 𝛼 ≠ 0 one has 𝜆 𝐶 󵄨󵄨 󵄨󵄨 󵄨 󵄨 𝑧} 𝑝(𝛼𝑥) = inf{𝜆 : |𝛼𝑥| ∧ 𝐶 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 ≤ 𝜆𝑧} = inf{𝜆 : |𝑥| ∧ 󵄨𝜑󵄨 ≤ |𝛼| 󵄨 󵄨 |𝛼| 𝐶 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󸀠 = inf{|𝛼| 𝜆󸀠 : |𝑥| ∧ 󵄨𝜑󵄨 ≤ 𝜆 𝑧} = |𝛼| inf{𝜆 : |𝑥| ∧ 𝐶 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 ≤ 𝜆𝑧} |𝛼| 󵄨 󵄨 = |𝛼| 𝑝(𝑥). 󵄨 󵄨 Any such seminorm is obviously a Riesz seminorm on 𝐸, i. e., |𝑥| ≤ 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 implies 𝑝(𝑥) ≤ 𝑝(𝑦) (cf. with (2.3)). The collection of all seminorms defines a locally convex topology on 𝐸 with a fundamental neighborhood system of zero⁵ consisting of solid sets. Below is a list of some properties of the introduced seminorms and the topology generated by means of them. 󵄨 󵄨 (1) The Archimedean principle yields |𝑥| ∧ 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 > 0 󳨐⇒ 𝑝(𝑥) > 0. (2) If in 𝐸 a complete system of finite elements exists, then the locally convex topology which is defined on 𝐸 by this system is Hausdorff. Indeed, for any 0 ≠ 𝑥 ∈ 𝐸, there 󵄨 󵄨 is a finite element 𝜑0 in that system such that |𝑥| ∧ 󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨 > 0. Hence the statement follows from the previous implication.

5 Sometimes called a neighborhood base of zero.

22 | 3 Finite, totally finite and selfmajorizing elements (3) If 𝐸 contains a countable complete set of finite elements, then the corresponding topology is metrizable. (4) If a discrete functional 𝑓 (i. e., lattice homomorphism from 𝐸 to ℝ; see Definition 5.6) does not vanish at the finite element 𝜑0 , then 𝑓 is continuous with respect to the generated seminorm. This can be easily seen from the estimation 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨 ≤ 𝑓(|𝑥|) = 𝑓 (|𝑥| ∧ 𝐶 󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨) ≤ 𝑝(𝑥) 𝑓(𝑧), which holds for sufficiently large 𝐶 > 0. (5) If the set Δ(𝐸) of all discrete functionals is total on 𝐸 (i. e., 𝑓(𝑥) = 0 for all 𝑓 ∈ Δ(𝐸) implies 𝑥 = 0), and if the vector lattice 𝐸 possesses a sufficient set of finite elements (see Section 5.3), then the topology is Hausdorff, and each discrete functional is continuous. Indeed, Property 2 implies that the topology is Hausdorff, since by Corollary 5.12, a sufficient set is also complete. The continuity of the discrete functionals follows from Property 4. Obviously, the set Φ1 (𝐸) is an ideal in 𝐸, i. e., a solid linear subspace of 𝐸. One now asks for stronger properties of the collection Φ1 (𝐸). The trivial cases for Φ1 (𝐸) to be a projection band in 𝐸 are Φ1 (𝐸) = 𝐸, and Φ1 (𝐸) = {0}. The general case is considered in the next theorem. Theorem 3.2 ([37, Theorem 2.13]). The ideal Φ1 (𝐸) is a projection band of the vector lattice 𝐸 if and only if 𝐸 = 𝐸1 ⊕ 𝐸0 , where Φ1 (𝐸1 ) = 𝐸1 and Φ1 (𝐸0 ) = {0}. In this case 𝐸1 = Φ1 (𝐸). Proof. If 𝐵 = Φ1 (𝐸) is a projection band in 𝐸, and 𝑃𝐵 the band projection onto 𝐵, then 𝐸 = 𝐵 ⊕ 𝐵0 , where 𝐵0 = 𝐵⊥ and Φ1 (𝐸) = Φ1 (𝐵) ⊕ Φ1 (𝐵0 ). Then Φ1 (𝐵) = 𝑃𝐵 Φ1 (𝐸) = Φ1 (𝐸) ∩ 𝐵 = Φ1 (𝐸), and therefore Φ1 (𝐵0 ) = {0}. If 𝐸 = 𝐵 ⊕ 𝐵0 with Φ1 (𝐵) = 𝐵 and Φ1 (𝐵0 ) = {0}, then Φ1 (𝐸) = Φ1 (𝐵) ⊕ Φ1 (𝐵0 ) = Φ1 (𝐵) = 𝐵. Notice that, obviously, the assertion of the theorem holds if Φ1 (𝐸) is a band in a Dedekind complete vector lattice 𝐸. The characterization of Φ1 (𝐸) as a band in an arbitrary Archimedean vector lattice is still open. Definition 3.3. A finite element 𝜑 ∈ 𝐸 is called totally finite if it has an 𝐸-majorant 𝑧 belonging to Φ1 (𝐸). The set of all totally finite elements of a vector lattice 𝐸 is also an ideal which will be denoted by Φ2 (𝐸). Obviously, the inclusions {0} ⊆ Φ2 (𝐸) ⊆ Φ1 (𝐸) ⊆ 𝐸 hold, which might be proper (see also Section 6.2). In general, if 𝐸 = 𝐶(𝑄), where the topological space 𝑄 is not compact, then K(𝑄) = Φ1 (𝐸) = Φ2 (𝐸) ≠ 𝐸. For 𝐸 = 𝐶[0, 1] there holds Φ2 (𝐸) = Φ1 (𝐸) = 𝐸.

3.1 Finite and totally finite elements in vector lattices

|

23

Later on, in Section 3.4, still another kind of finite element will be studied, namely the selfmajorizing elements, each of which has its modulus as a majorant; see Definition 3.35. The sets of totally finite and selfmajorizing elements in general turn out to be different from Φ1 (𝐸). The next example shows that vector lattices with {0} ≠ Φ2 (𝐸) ≠ Φ1 (𝐸) exist. Example 3.4 (Kaplansky vector lattice). This vector lattice provides a vector lattice 𝐸 with Φ1 (𝐸) ≠ Φ2 (𝐸) and, after a slight modification of the construction, we get an example of a vector lattice 𝐸 with {0} = Φ1 (𝐸) ≠ 𝐸 mentioned earlier in this section as case (iii). Both examples will also be of use several times later on (especially in Sections 6.2 and 6.4) for constructing counterexamples (see Example 6.39) which show that the conditions posed in Theorem 6.32 are essential. Let 𝑇 = [−2, 2] \ {1, 12 , 13 , ...}. The Kaplansky vector lattice (see [25, XV.3],[116, 117]), which will be denoted by K, consists of all functions 𝑥 on [−2, 2] restricted to 𝑇 such that – 𝑥 is continuous on [−2, 2] except at a finite number of points 1𝑛 ; 󵄨 󵄨 – for any 𝑛 ∈ ℕ the finite limit lim1 󵄨󵄨󵄨𝑡 − 𝑛1 󵄨󵄨󵄨𝑥 (𝑡) exists. 𝑡→ 𝑛

The functions

𝜈

𝑒𝜈(𝑡) = ∑ 𝑘=1

1 , |𝑘𝑡 − 1|

𝑡 ∈ 𝑇,

𝜈 = 1, 2, . . .

(3.3)

1 ≥ 1 on belong to K and satisfy there the condition (Σ󸀠 ). Moreover, one has 𝑒𝜈 (𝑡) ≥ |𝑡−1| [0, 1] ∩ 𝑇. It is now easy to show that K is a uniformly complete vector lattice of type (Σ) with the property Φ1 (K) ≠ Φ2 (K). Indeed, in order to see this, we list some properties of the finite and totally finite elements in K. (a) If 𝜑 is a finite element in K then 𝜑(0) = 0. 󵄨 󵄨 Indeed, by assuming 𝜑(0) ≠ 0, there is a number 𝛿 > 0 such that 󵄨󵄨󵄨𝜑(𝑡)󵄨󵄨󵄨 > 0 for all 𝑡 ∈ 𝑈 = (−𝛿, 𝛿) ∩ 𝑇. If 𝑒𝑚 is a majorant of the finite element 𝜑, then choose a natural number 𝜈 with the properties 𝜈1 < 𝛿 and 𝜈 > 𝑚. Due to the finiteness of 𝜑, 󵄨 󵄨 it holds that 𝑒𝜈 ∧ 𝑛 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 ≤ 𝑐𝜈 𝑒𝑚 for all 𝑛 ∈ ℕ and some 𝑐𝜈 . In particular, one has

𝑒𝜈(𝑡) ≤ 𝑐𝜈 𝑒𝑚 (𝑡) for 𝑡 ∈ 𝑈. The last inequality, however, is impossible due to the choice of 𝜈. (b) If 𝜑 ∈ K and a real 𝛿 > 0 exists such that 𝜑(𝑡) = 0 for all 𝑡 ∈ [0, 𝛿)∩𝑇, then 𝜑 ∈ Φ1 (K). 󵄨 󵄨 Indeed, for 𝜑 ∈ K, there is a number 𝜈 such that 𝜈1 < 𝛿 and 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 ≤ 𝜆 𝜑 𝑒𝜈 . For 𝑚 ≤ 𝜈 󵄨 󵄨 the estimation 𝑒𝑚 ∧ 𝑛 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 ≤ 𝑒𝜈 obviously holds for all 𝑛 ∈ ℕ. 󵄨 󵄨 If 𝑡 ∈ [0, 𝛿) ∩ 𝑇, then 𝜑(𝑡) = 0 and (𝑒𝑚 ∧ 𝑛 󵄨󵄨󵄨𝜑󵄨󵄨󵄨)(𝑡) = 0 for each 𝑚, 𝑛 ∈ ℕ and 𝑡 ∈ [0, 𝛿). 𝑚 1 1 If 𝑡 ∈ [𝛿, 2] ∩ 𝑇 and 𝑚 > 𝜈, then 𝑒𝑚 (𝑡) = 𝑒𝜈 (𝑡) + ∑𝑚 𝑘=𝜈+1 |𝑘𝑡−1| , where ∑𝑘=𝜈+1 |𝑘𝑡−1| is a 1 continuous function on the compact interval [𝛿, 2]. The latter implies ∑𝑚 𝑘=𝜈+1 |𝑘𝑡−1| ≤

24 | 3 Finite, totally finite and selfmajorizing elements 𝛽𝑚 for some 𝛽𝑚 > 0. One then has 𝑚

󵄨 󵄨 󵄨 󵄨 (𝑒𝑚 ∧ 𝑛 󵄨󵄨󵄨𝜑󵄨󵄨󵄨)(𝑡) ≤ (𝑒𝜈 ∧ 𝑛 󵄨󵄨󵄨𝜑󵄨󵄨󵄨)(𝑡) + (( ∑ 𝑘=𝜈+1

1 󵄨 󵄨 ) ∧ 𝑛 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 )(𝑡) |𝑘𝑡 − 1|

≤ 𝑒𝜈 (𝑡) + 𝛽𝑚 ≤ (1 + 𝛽𝑚 )𝑒𝜈 (𝑡) for all 𝑛 ∈ ℕ,

(3.4)

which shows that 𝜑 is a finite element and e𝜈 is one of its majorants. (c) The element 𝜓 ∈ K is totally finite, i. e., 𝜓 ∈ Φ2 (K), if and only if 𝛿 > 0 exists such that 𝜓(𝑡) = 0 for all 𝑡 ∈ (−𝛿, 𝛿) ∩ 𝑇. For necessity let 𝜓 ∈ Φ2 (K), 𝜓 ≥ 0, and let 𝜑 ∈ Φ1 (K) be a majorant for 𝜓. If we assume that for some sequence (𝑡𝑘 )𝑘∈ℕ ⊂ 𝑇 with 𝑡𝑘 󳨀→ 0 there is 𝜓(𝑡𝑘) > 0 for 𝑘→∞

𝑘 = 1, 2, . . . , then from the inequality 1 ∧ 𝑛𝜓(𝑡) ≤ 𝑐1 𝜑(𝑡),

𝑡 ∈ 𝑇, 𝑛 ∈ ℕ

there would follow 1 ≤ 𝑐1 𝜑(𝑡𝑘 ) for 𝑘 = 1, 2, . . . . This contradicts 𝜑(𝑡𝑘) 󳨀→ 𝜑(0) 𝑘→∞

since 𝜑(0) = 0, as was established in (a). For the sufficiency of the condition, consider a function 𝜓 ∈ K such that 𝜓(𝑡) = 0 for all 𝑡 ∈ (−𝛿, 𝛿) ∩ 𝑇 for some 𝛿 > 0. There is a number 𝜈 such that 1𝜈 < 𝛿 and 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨𝜓󵄨󵄨 ≤ 𝜆 𝜓 𝑒𝜈 . Put 𝛼 = max𝑡∈[−2,0] 󵄨󵄨𝜓󵄨󵄨. The function 𝛼, { { { { { { − 𝛼𝛿 𝑡 , { { { 𝜑(𝑡) = { 0, { { { { 𝑒𝜈(𝛿)( 2𝛿 𝑡 − 1), { { { { 𝑒𝜈 (𝑡), {

𝑡 ∈ [−2, −𝛿) 𝑡 ∈ [−𝛿, 0) 𝑡 ∈ [0, 𝛿2 ) ∩ 𝑇 𝑡 ∈ [ 𝛿2 , 𝛿) ∩ 𝑇 𝑡 ∈ [𝛿, 2] ∩ 𝑇

belongs to Φ1 (K), according to (b). We show that 𝜑 is a majorant for 𝜓. If 𝑡 ∈ [−2, −𝛿), then by continuity 𝑒𝑚 (𝑡) ≤ 𝑘𝑚 𝛼 for some 𝑘𝑚 > 0 and 󵄨 󵄨 (𝑒𝑚 ∧ 𝑛 󵄨󵄨󵄨𝜓󵄨󵄨󵄨)(𝑡) ≤ 𝑒𝑚 (𝑡) ≤ 𝑘𝑚 𝛼 = 𝑘𝑚 𝜑(𝑡). 󵄨 󵄨 If 𝑡 ∈ [−𝛿, 𝛿] ∩ 𝑇, then (𝑒𝑚 ∧ 𝑛 󵄨󵄨󵄨𝜓󵄨󵄨󵄨)(𝑡) = 0, and so 󵄨 󵄨 (𝑒𝑚 ∧ 𝑛 󵄨󵄨󵄨𝜓󵄨󵄨󵄨)(𝑡) ≤ {

− 𝛼𝛿 𝑡 = 𝜑(𝑡) , 𝜑(𝑡) ,

𝑡 ∈ [−𝛿, 0] 𝑡 ∈ (0, 𝛿] ∩ 𝑇. 𝑚

1 If 𝑡 ∈ (𝛿, 2] ∩ 𝑇, we may assume 𝑚 > 𝜈. Then, as in (b), we have ∑𝑘=𝜈+1 |𝑘𝑡−1| ≤ 𝛽𝑚 for 󵄨󵄨 󵄨󵄨 1 some 𝛽𝑚 > 0, since 𝜈 < 𝛿. Now (𝑒𝑚 ∧ 𝑛 󵄨󵄨𝜓󵄨󵄨)(𝑡) is estimated on (𝛿, 2] ∩ 𝑇 in the same way as in (3.4), where, due to 𝜑(𝑡) = 𝑒𝜈 (𝑡), the upper bound is (1 + 𝛽𝑚 )𝜑(𝑡). Now we have an example for the relation 𝐸 ≠ Φ1 (𝐸) ≠ Φ2 (𝐸) ≠ {0} to hold, e. g., the function 𝑡− = max{−𝑡, 0} belongs to Φ1 (𝐸), but not to Φ2 (𝐸).

3.1 Finite and totally finite elements in vector lattices

| 25

Example 3.5. A vector lattice 𝐸 of type (Σ) with {0} = Φ2 (𝐸) = Φ1 (𝐸) ≠ 𝐸. Let {𝑟𝑚 : 𝑚 ∈ ℕ} be the set of all rational numbers in 𝑇 = [0, 1]. The vector lattice⁶ 𝐸 = 𝐸(𝑇) consists of all functions 𝑥 on 𝑇, each of which can be represented as 𝑁

𝑥(𝑡) = 𝛼(𝑡) + ∑ 𝑚=1

𝛼𝑚 , |𝑡−𝑟𝑚 |

where 𝛼(𝑡) is some continuous function, 𝛼1 , 𝛼2 , . . . , 𝛼𝑚 are real numbers, and 𝑁 = 𝑁(𝑥) a natural. It is clear that each function 𝑥 ∈ 𝐸 has a finite limit lim |𝑡 − 𝑟𝑚 |𝑥(𝑡) at each 𝑡→𝑟𝑚

point 𝑟𝑚 ∈ ℚ ∩ 𝑇. The value of 𝑥 at the points 𝑟𝑚 for 𝑚 = 1, . . . , 𝑁 is assumed to be +∞, −∞, 0 if the coefficient 𝛼𝑚 is > 0, < 0, 0 respectively. Observe that the functions 𝜈

𝑒𝜈 (𝑡) = 1 + ∑ 𝑚=1

1 , |𝑡−𝑟𝑚 |

𝜈 = 1, 2, . . .

belong to 𝐸 and satisfy the condition (Σ󸀠 ). Thus, 𝐸 is a vector lattice of type (Σ). We show that except for the zero-function, no element can be finite in 𝐸. Indeed, let 𝜑 ∈ 𝐸, 𝜑 > 0. Then 𝜑(𝑡0 ) > 0 at some point 𝑡0 and 𝜑(𝑡) > 0 holds also in some neighborhood 𝑈 of 𝑡0 . We make sure that 𝜑 cannot be a finite element in the vector lattice 𝐸. By way of contradiction, assume that there is a number 𝜈0 such that for any 𝜈 there is a number 𝑐𝜈 with the property 𝑒𝜈 ∧ 𝑛𝜑 ≤ 𝑐𝜈 𝑒𝜈0

for all 𝑛 ∈ ℕ.

In the neighborhood 𝑈 there is a rational number 𝑟𝑚0 with 𝑚0 > 𝜈0 and 𝑒𝜈0 (𝑟𝑚0 ) < ∞. If 𝑛 is sufficiently large, then one has 𝑒𝑚0 (𝑟𝑚0 ) = ∞ , and therefore (𝑒𝑚0 ∧ 𝑛𝜑)(𝑟𝑚0 ) = 𝑛𝜑(𝑟𝑚0 ). Due to large 𝑛, the last value might be arbitrarily large. This contradiction shows that the element 𝜑 is not finite. Now it is easy to see that also Φ2 (𝐸) = {0}. Thus arises the natural problem of describing all, or at least some, finite elements in various vector lattices of sequences, functions, operators, etc., as already started after Definition 3.1. The investigation of finite elements, especially in Banach lattices, gives some additional information on the inner structure of such spaces and might be used to discover further interesting properties. Finite and totally finite elements in vector lattices have been thoroughly studied in a series of papers (see [36–38, 89– 92, 131]). When the finiteness of some class of elements has to be proved in a particular vector lattice, it will be clear that special techniques have to be developed in order to establish Formula (3.1) contained in Definition 3.1. Moreover, an analysis of Formula (3.1) will help to derive more information about the structure of the finite elements in many

6 𝐸(𝑇) is constructed similarly to the Kaplansky vector lattice.

26 | 3 Finite, totally finite and selfmajorizing elements special cases. Some typical results are contained in the theorems below and in the further chapters of the book. As a rule, an additional structure of the vector lattice will give some more, and sometimes even exhaustive, information about its finite elements. From the definitions one has immediately the following theorem. Theorem 3.6. If a vector lattice 𝐸 has an order unit, then Φ1 (𝐸) = Φ2 (𝐸) = 𝐸. Proof. Indeed, if 1 is an order unit in 𝐸, then for each 𝑥 ∈ 𝐸 there is positive number 󵄨 󵄨 𝑐𝑥 , such that |𝑥| ≤ 𝑐𝑥 1. For any 𝑦 ∈ 𝐸 one has |𝑥| ∧ 𝑛 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ≤ 𝑐𝑥 1 which, by Definition 3.1, shows that the element 𝑦 is finite. As a consequence, for the following classical vector lattices, one can immediately see that any element is finite. Corollary 3.7. If 𝐸 is one of the vector lattices c, ℓ∞ , 𝐶(𝐾) for a compact Hausdorff space 𝐾, 𝐿 ∞ (𝜇), then Φ1 (𝐸) = Φ2 (𝐸) = 𝐸. It was shown in [90] that any element 𝜑 ∈ Φ2 (𝐸) possesses an 𝐸-majorant which itself is a totally finite element; see Theorem 6.15 below. This means that there is no further specification in this direction. It is clear that Φ1 (𝐸) = 𝐸 implies Φ2 (𝐸) = Φ1 (𝐸). Indeed, if 𝜑 ∈ Φ1 (𝐸), then there is a majorant in 𝐸, which, due to 𝐸 = Φ1 (𝐸), is a finite element, and 𝜑 is therefore totally finite. Proposition 3.8. Each atom in a vector lattice is a totally finite element with itself as a majorant⁷. Proof. If 𝑎 is an atom of 𝐸, then for each 𝑥 ∈ 𝐸+ define 𝜆 𝑎 (𝑥) = sup{𝑟 ∈ ℝ+ : 𝑟𝑎 ≤ 𝑥}. As 𝐸 is Archimedean, 𝜆 𝑎 (𝑥) < ∞, for all 𝑥 ∈ 𝐸+ . For 𝑥 ∈ 𝐸 and every 𝑛, one has |𝑥| ∧ 𝑛𝑎 ≤ 𝑛𝑎, i. e., 1𝑛 (|𝑥| ∧ 𝑛𝑎) ≤ 𝑎 and, by taking into consideration that 𝑎 is an atom, one obtains 1𝑛 (|𝑥| ∧ 𝑛𝑎) = 𝑟𝑛󸀠 𝑎, and therefore |𝑥| ∧ 𝑛𝑎 = 𝑟𝑛𝑎 ≤ |𝑥| for some 𝑟𝑛 , which gives |𝑥| ∧ 𝑛𝑎 ≤ 𝜆 𝑎 (|𝑥|)𝑎. Therefore, 𝑎 is finite with itself as an 𝐸-majorant. Let 𝑋 ⊂ 𝐸 be a vector sublattice of the vector lattice 𝐸. An element 𝑧 ∈ 𝐸+ is called a generalized order unit for 𝑋 if for each 𝑥 ∈ 𝑋 there is a real number 𝐶𝑥 with |𝑥| ≤ 𝐶𝑥 𝑧. Note that then 𝑋 belongs to the ideal generated (in E) by the element 𝑧, and that 𝑧 is not required to belong to 𝑋+ = 𝑋 ∩ 𝐸+ . Theorem 3.9. Let 𝐸 be a vector lattice. If 𝜑 ∈ 𝐸 is a finite element, then {𝜑}⊥⊥ has a generalized order unit and {𝜑}⊥⊥ ⊂ Φ1 (𝐸). Proof. If 𝜑 ∈ 𝐸 is finite, then 𝑧 ∈ 𝐸+ exists such that for each 𝑥 ∈ 𝐸 there is a real number 𝑐𝑥 > 0 with |𝑥| ∧ (𝑛|𝜑|) ≤ 𝑐𝑥 𝑧 for all 𝑛 ∈ ℕ.

7 Later termed a selfmajorizing element.

3.1 Finite and totally finite elements in vector lattices

|

27

It follows from Theorem 3.4 of [9] that |𝑥| = sup{|𝑥| ∧ 𝑛|𝜑|} ≤ 𝑐𝑥 𝑧

for all

𝑥 ∈ {𝜑}⊥⊥ ,

which implies that the element 𝑧 is a generalized order unit of {𝜑}⊥⊥ . Now for each 𝑣 ∈ {𝜑}⊥⊥ and arbitrary 𝑥 ∈ 𝐸, it is clear that |𝑥| ∧ 𝑛|𝑣| ∈ {𝜑}⊥⊥ and, again by the same Theorem 3.4, one has |𝑥| ∧ 𝑛|𝑣| = sup{|𝑥| ∧ 𝑛|𝑣| ∧ 𝑚|𝜑| : ∀𝑚 ∈ ℕ} = sup{(|𝑥| ∧ 𝑚|𝜑|) ∧ 𝑛|𝑣| : ∀𝑚 ∈ ℕ} ≤ (𝑐𝑥 𝑧) ∧ (𝑛|𝑣|) ≤ 𝑐𝑥 𝑧 for all 𝑛. So 𝑣 is finite, and {𝜑}⊥⊥ ⊂ Φ1 (𝐸). The next result shows that the converse of Theorem 3.6 is true whenever the vector lattice 𝐸 has a weak order unit. Corollary 3.10. Let 𝐸 be a vector lattice with a weak order unit. Then Φ1 (𝐸) = Φ2 (𝐸) = 𝐸 if and only if 𝐸 has an order unit. Proof. Indeed. We only have to prove the existence of an order unit if 𝐸 has a weak order unit and Φ1 (𝐸) = Φ2 (𝐸) = 𝐸. If 𝑒 is a weak order unit in 𝐸 = Φ1 (𝐸), then by the theorem {𝑒}⊥⊥ has a generalized order unit which, due to {𝑒}⊥⊥ = 𝐸, is obviously an order unit. Note that we have established that any generalized order unit of the band generated by the weak order unit serves as an order unit in 𝐸. However, a weak order unit of a vector lattice 𝐸 fails to be an order unit, in general, even if Φ1 (𝐸) = Φ2 (𝐸) = 𝐸 and 𝐸 has an order unit. For example, if 𝐸 = c and 𝑢 = (1, 12 , . . . , 1𝑛 , . . .), then 𝑢 is a weak order unit of 𝐸, but not an order unit. The next result is a characterization of finite elements in vector lattices with (𝑝𝑝𝑝). The equivalence (1) ⇐⇒ (3) in case of a Dedekind complete vector lattice was proved in [130, Theorem 1]. Theorem 3.11. Let 𝐸 be a vector lattice with the principal projection property (𝑝𝑝𝑝) and 𝜑 ∈ 𝐸. Then the following statements are equivalent: (1) 𝜑 is a finite element of 𝐸; (2) {𝜑}⊥⊥ has a generalized order unit 𝑧 ∈ 𝐸+ ; (3) {𝜑}⊥⊥ has an order unit 𝑧0 ∈ {𝜑}⊥⊥ . Proof. (1) ⇒ (2) is precisely Theorem 3.9. (2) ⇒ (3): If 𝑧 ∈ 𝐸+ is a generalized order unit of {𝜑}⊥⊥ , then for each 𝑥 ∈ {𝜑}⊥⊥ , there is a positive number 𝑐𝑥 such that |𝑥| ≤ 𝑐𝑥 𝑧. Let 𝑃𝜑 be the band projection from 𝐸 onto {𝜑}⊥⊥ . Then, |𝑥| = 𝑃𝜑 |𝑥| ≤ 𝑃𝜑 (𝑐𝑥 𝑧) = 𝑐𝑥𝑃𝜑 𝑧 = 𝑐𝑥 𝑧0 , where 𝑧0 = 𝑃𝜑 𝑧 ∈ {𝜑}⊥⊥ . This implies that 𝑧0 is an order unit of {𝜑}⊥⊥ .

28 | 3 Finite, totally finite and selfmajorizing elements (3) ⇒ (1): If 𝑧0 is an order unit of {𝜑}⊥⊥ , then, due to 𝑃𝜑 |𝑥| ∈ {𝜑}⊥⊥ , for arbitrary 𝑥 ∈ 𝐸, there is a positive number 𝑐𝑥 such that 𝑃𝜑 |𝑥| ≤ 𝑐𝑥 𝑧0 . Therefore, |𝑥| ∧ 𝑛|𝜑| ≤ sup{|𝑥| ∧ 𝑛|𝜑|} = 𝑃𝜑 |𝑥| ≤ 𝑐𝑥 𝑧0

for all 𝑛 ∈ ℕ,

and 𝜑 is a finite element of 𝐸. Remark 3.12. (1) It has been proved a little more. Namely, if 𝜑 is a finite element and 𝑧 an arbitrary one of its generalized order units, then 𝑃𝜑 𝑧 is an order unit in {𝜑}⊥⊥ . If, in addition, {𝜑}⊥⊥ ⊆ Φ1 (𝐸), then {𝜑}⊥⊥ ⊆ Φ2 (𝐸). (2) The proof also shows that the finiteness of an element in an arbitrary Archimedean vector lattice can be detected by the properties of its principal band: let 𝐸 be an arbitrary Archimedean vector lattice and let the element 𝜑 ∈ 𝐸 be such that {𝜑}⊥⊥ is a projection band. Then 𝜑 is a finite element (with the majorant 𝑧) if and only if {𝜑}⊥⊥ contains an order unit (namely, 𝑃𝜑 𝑧). In particular, if 𝐸 is a 𝜎-Dedekind complete vector lattice, then Φ1 (𝐸) = 𝐸 if and only if each principal band possesses an order unit. (3) If 𝐸 is a vector lattice with (𝑝𝑝𝑝) and the element 𝜑 ∈ 𝐸 is finite, then for each 𝑥 ∈ 𝐸 the supremum exists on the left-hand side of Formula (3.1), which is the projection 𝑃𝜑 (|𝑥|) on the band {𝜑}⊥⊥ ; see Formula (2.2). All these projections lie in the ideal generated by the element 𝑧, i. e., 𝑃𝜑 (𝑥+ ), 𝑃𝜑 (𝑥− ), 𝑃𝜑 (𝑥), 𝑃𝜑 (|𝑥|) ∈ 𝐼𝑧 . Combining the Theorems 3.9 and 3.11 we have the following theorem. Theorem 3.13. Let 𝐸 be a vector lattice with the principle projection property (𝑝𝑝𝑝). Then Φ1 (𝐸) = Φ2 (𝐸), and Φ1 (𝐸) also has the (𝑝𝑝𝑝). Proof. Let 𝜑 ∈ Φ1 (𝐸), and let 𝑧 be its majorant. Then 𝑃𝜑 𝑧 is also a majorant for 𝜑. Due to Theorem 3.9, one has {𝜑}⊥⊥ ⊂ Φ1 (𝐸) and gets Φ1 (𝐸) = Φ2 (𝐸). If 𝑣 ∈ Φ1 (𝐸), then {𝑣}⊥⊥ is a projection band in 𝐸, which is a subset of Φ1 (𝐸). This shows that the ideal Φ1 (𝐸) has (𝑝𝑝𝑝). It is easy to show that Φ1 (𝐸) = Φ2 (𝐸) = 𝐸 holds, both for the vector lattice 𝐸 = c00 of all sequences with finite support (see also case (i) on p. 20), and also for the vector lattice 𝐸 = K(𝑄) of all finite continuous functions on the locally compact topological Hausdorff space 𝑄. Another result for a vector lattice 𝐸 to satisfy Φ1 (𝐸) = 𝐸 uses the structure of a strict inductive limit (see [89, Theorem 5.3]). This result will be proved later in Section 9.3 in connection with the representation of 𝐿𝐹-vector lattices. The finiteness of elements in vector lattices is an order isomorphic property, i. e., if two vector lattices 𝐸 and 𝐹 are lattice isomorphic by means of a Riesz isomorphism 𝑇, then 𝜑 ∈ 𝐸 is finite if and only if 𝑇𝜑 is finite in 𝐹. For the sake of convenience we state this property in the following proposition without proof.

3.2 Finite elements in Banach lattices

|

29

Proposition 3.14. Let 𝐸 and 𝐹 be vector lattices and 𝑇 : 𝐸 → 𝐹 a Riesz isomorphism. Then 𝑇 (Φ1 (𝐸)) = Φ1 (𝑇(𝐸)) = Φ1 (𝐹), and 𝑇 (Φ2 (𝐸)) = Φ2 (𝐹). It is worth pointing out that if 𝑇 : 𝐸 → 𝐹 is not a lattice isomorphism, then for 𝜑 ∈ Φ1 (𝐸), the element 𝑇𝜑 may fail to be finite in 𝐹, even if 𝑇 is an interval-preserving⁸ (linear) lattice homomorphism, i. e., satisfies the condition 𝑇(𝑥 ∧ 𝑦) = 𝑇𝑥 ∧ 𝑇𝑦 for all 𝑥, 𝑦 ∈ 𝐸. For example, let 𝐸 = ℓ∞ , 𝐹 = c0 , and 𝑇 : 𝐸 → 𝐹 defined by 𝑇(𝜆 𝑛) = (𝑡𝑛𝜆 𝑛) for (𝜆 𝑛) ∈ 𝐸, where 0 < 𝑡𝑛 ∈ ℝ and 𝑡𝑛 → 0. It is easy to see that 𝑇 is an interval-preserving lattice homomorphism but not a lattice isomorphism. Although Φ1 (𝐸) = 𝐸 (see Corollary 3.7), the element 𝑇𝑒 = (𝑡𝑛) is not finite in 𝐹, where 𝑒 is the sequence of 𝐸 with all members equal to 1. The latter will be clear and is easily obtained as a consequence of Theorem 3.18; see p. 32.

3.2 Finite elements in Banach lattices If the underlying vector lattice 𝐸 is a Banach lattice, then the finite elements can be characterized similarly to Theorem 3.11 regardless of the property (𝑝𝑝𝑝). Theorem 3.15. Let 𝐸 be a Banach lattice, and 𝜑 ∈ 𝐸. Then the following statements are equivalent: (1) 𝜑 is a finite element; (2) the closed unit ball 𝐵({𝜑}⊥⊥) of {𝜑}⊥⊥ is order bounded in 𝐸: (3) {𝜑}⊥⊥ has a generalized order unit. Proof. We show first the equivalence of (1) and (3) In view of Theorem 3.9 it has only to be shown that the element 𝜑 is finite if {𝜑}⊥⊥ has a generalized order unit. In fact, let 𝑧 ∈ 𝐸+ be a generalized order unit of {𝜑}⊥⊥ . Define a norm on {𝜑}⊥⊥ by ‖𝑥‖𝑧 = inf{𝜆 > 0 : |𝑥| ≤ 𝜆𝑧},

𝑥 ∈ {𝜑}⊥⊥ .

Then by Theorem 12.20 in [9], the space ({𝜑}⊥⊥ , ‖ ⋅ ‖𝑧 ) is an 𝐴𝑀-space, where |𝑥| ≤ ‖𝑥‖𝑧 𝑧 holds. Since the band {𝜑}⊥⊥ is closed in 𝐸 ([95, Proposition 1.2.3]) ({𝜑}⊥⊥, ‖ ⋅ ‖) also is a Banach space. The open mapping theorem implies that the norms ‖ ⋅ ‖ and ‖ ⋅ ‖𝑧 are equivalent on {𝜑}⊥⊥ . In particular, there is a 𝐶 > 0 such that ‖𝑥‖𝑧 ≤ 𝐶‖𝑥‖ for all 𝑥 ∈ {𝜑}⊥⊥ . Now |𝑥| ≤ ‖𝑥‖𝑧 𝑧 for each 𝑥 ∈ {𝜑}⊥⊥ , implies ‖𝑥‖𝑧 ≤ 𝐶, i. e., |𝑥| ≤ 𝐶𝑧 for each 𝑥 ∈ {𝜑}⊥⊥ with ‖𝑥‖ ≤ 1. If 𝑥 ∈ 𝐸 is now an arbitrary element, then 0 ≤ |𝑥| ∧ 𝑛|𝜑| ≤ |𝑥| (and hence ‖|𝑥| ∧ 𝑛|𝜑|‖ ≤ ‖𝑥‖) implies |𝑥| ∧ 𝑛|𝜑| ≤ ‖𝑥‖𝐶𝑧 for all 𝑛 ∈ ℕ, which means that 𝜑 is finite.

8 That is, 𝑇[0, 𝑥] = [0, 𝑇𝑥], ∀𝑥 ∈ 𝐸+ . In the case of a Dedekind complete 𝐹, the operator 𝑇 is called a Maharam operator; see [77, § 3.4].

30 | 3 Finite, totally finite and selfmajorizing elements (2) ⇒ (3) If the ball 𝐵({𝜑}⊥⊥ ) is order bounded⁹ in 𝐸, then there is an 𝑧 ∈ 𝐸+ such that 𝐵({𝜑}⊥⊥ ) ⊂ [−𝑧, 𝑧]. So, it is clear that 𝑧 is a generalized order unit for {𝜑}⊥⊥ . (3) ⇒ (2) In the first part of the proof it was established that any norm bounded subset of {𝜑}⊥⊥ is order bounded. Remark 3.16. (1) In the proof we have established the following fact: since in {𝜑}⊥⊥ the two norms ‖ ⋅ ‖ and ‖ ⋅ ‖𝑧 are equivalent, then any ‖ ⋅ ‖-bounded set 𝐴 ⊂ {𝜑}⊥⊥ is also order bounded. (2) The principal band generated by a finite element may fail to possess an order unit as the following example shows (see [36]). Let 𝐸 = 𝐶[0, 1], and 𝐻𝑐 = {𝑥 ∈ 𝐸 : 𝑥(𝑡) = 0, ∀𝑡 ∈ [0, 𝑐]} for 𝑐 ∈ (0, 1). Then (i) Φ1 (𝐸) = 𝐸. (ii) 𝐻𝑐 is a principal band for each 𝑐 ∈ (0, 1). Moreover, 𝐻𝑐 = {𝜑}⊥⊥ for any 𝜑 ∈ 𝐻𝑐 satisfying 𝜑(𝑡) ≠ 0 for 𝑡 ∈ (𝑐, 1]. (iii) 𝐻𝑐 does not possess any order unit. However, each function 𝑧 ∈ 𝐸 with 𝑧(𝑡) > 0 for 𝑡 ∈ (𝑐 − 𝛿, 1] is a generalized order unit, where 𝛿 is some positive number. (3) If 𝐸 is only a normed vector lattice without norm completeness (i. e., 𝐸 is not a Banach lattice), then the order boundedness of 𝐵({𝜑}⊥⊥ ) yields that the element 𝜑 is finite, but the converse statement, in general, is false as the following example shows. Let 0 < 𝜑 ∈ 𝐿 1 (𝑎, 𝑏), and take the order ideal generated by 𝜑 in 𝐿 1 (𝑎, 𝑏) 𝐸 = 𝐼𝜑 = { 𝑥 ∈ 𝐿 1 (𝑎, 𝑏) : ∃𝜆 > 0 with |𝑥| ≤ 𝜆|𝜑| }, then (𝐸, ‖ ⋅ ‖1 ), where ‖ ⋅ ‖1 is the integral-norm in 𝐿 1 (𝑎, 𝑏), is a normed vector lattice which fails to be a Banach lattice (cf. [144, Exercise 17.18]), and each of its elements is finite in 𝐸, as 𝐸 has an order unit. On the other hand, since Φ1 (𝐿 1 (𝑎, 𝑏)) = {0} (see (b) after Theorem 3.18 below), the closed unit ball of {𝜑}⊥⊥ is not order bounded in 𝐿 1 (𝑎, 𝑏). It follows that 𝐵({𝜑}⊥⊥ ) is not order bounded in 𝐸 although the norm on 𝐸 is even order continuous. The proof of the subsequent theorem requires some information about the set of all atoms of norm 1 in atomic Banach lattices which are collected in the next proposition (for more details see [34]). For a Banach lattice 𝐸 denote by Γ𝐸 the set of all atoms of 𝐸 with norm 1. It is not difficult to verify that Γ𝐸 consists of pairwise disjoint elements, hence Γ𝐸 is a linearly independent system. According to Proposition 3.8, it holds that Γ𝐸 ⊂ Φ1 (𝐸).

9 Each order interval [𝑎, 𝑏] is included in some symmetrical order interval [−𝑧, 𝑧]. Take 𝑧 = |𝑎| ∨ |𝑏|.

3.2 Finite elements in Banach lattices

|

31

Proposition 3.17. Let 𝐸 be an atomic Banach lattice. Then (1) 𝑥 = sup{𝜆 𝑎 (𝑥)𝑎: 𝑎 ∈ Γ𝐸 } for each 𝑥 ∈ 𝐸+ , where for 𝑥 ∈ 𝐸+ and 𝑎 ∈ Γ𝐸 , the number 𝜆 𝑎 (𝑥) is defined by 𝜆 𝑎 (𝑥) = sup{𝑟 ∈ ℝ+ : 𝑟𝑎 ≤ 𝑥}.

(3.5)

(2) if, in addition, 𝐸 has an order continuous norm, then 𝜆 𝑎 (𝑥) = 0 for all but countable many 𝑎 ∈ Γ𝐸 , and 𝑥 = ∑𝑎∈Γ𝐸 𝜆 𝑎 (𝑥)𝑎 for each 𝑥 ∈ 𝐸+ . Hence, 𝐸 = span(Γ𝐸 ), where the series is norm-convergent and span(Γ𝐸 ) denotes the set of all linear combinations 𝜆 1 𝑎1 + 𝜆 2 𝑎2 + ⋅ ⋅ ⋅ + 𝜆 𝑛𝑎𝑛 with 𝜆 𝑘 ∈ ℝ, 𝑎𝑘 ∈ Γ𝐸 , 𝑘 = 1, . . . , 𝑛 and 𝑛 ∈ ℕ. Proof. (1) It is clear that 𝑥 ≥ 𝜆 𝑎 (𝑥)𝑎 for any 𝑎 ∈ Γ𝐸 . If 𝑧 ≥ 𝜆 𝑎 (𝑥)𝑎 for all 𝑎 ∈ Γ𝐸 , then 𝑧∧𝑥 ≥ 𝜆 𝑎 (𝑥)𝑎 for all 𝑎 ∈ Γ𝐸 . If one would have 𝑥−𝑧∧𝑥 > 0 then (due to 𝐸 being atomic), 𝑏 ∈ Γ𝐸 and 𝑟 > 0 exist such that 0 < 𝑟𝑏 ≤ 𝑥 − 𝑧 ∧ 𝑥 and (𝑟 + 𝜆 𝑏 (𝑥))𝑏 ≤ 𝑥 − 𝑧 ∧ 𝑥 + 𝑧 ∧ 𝑥 = 𝑥. The definition of 𝜆 𝑏 (𝑥) now implies 𝑟 + 𝜆 𝑏 (𝑥) ≤ 𝜆 𝑏 (𝑥), a contradiction. This means 𝑥 = 𝑧 ∧ 𝑥 ≤ 𝑧 and thus the relation (3.5) is proved. (2) Due to the order continuity of the norm for 𝑥 ∈ 𝐸+ for any 𝑛 ∈ ℕ, the set Γ𝑛 = {𝑎 ∈ Γ𝐸 : 𝜆 𝑎 (𝑥) ≥ 𝑛1 } is finite . So {𝑎 ∈ Γ𝐸 : 𝜆 𝑎 (𝑥) > 0} = ⋃∞ 𝑛=1 Γ𝑛 is at most countable, say {𝑎1 , . . . , 𝑎𝑛 , . . .}. Then 𝜆 𝑎1 (𝑥)𝑎1 + ⋅ ⋅ ⋅ + 𝜆 𝑎𝑛 (𝑥)𝑎𝑛 ↑ 𝑥. Again, by order continuity of the norm we get 𝑥 = ∑∞ 𝑛=1 𝜆 𝑎𝑛 (𝑥)𝑎𝑛 . For the proof of the next result we also need the fact that in every 𝐴𝑀-space the lattice operations are weakly sequentially continuous (see p. 13), and use Theorem 2.8 on norm compactness of the order intervals in atomic Banach lattices with order continuous norm. Theorem 3.18. Let 𝐸 be a Banach lattice with an order continuous norm. Then (1) Φ1 (𝐸) = Φ2 (𝐸) = span(Γ𝐸 ); (2) Φ1 (𝐸) is closed in 𝐸 if and only if Γ𝐸 is a finite set. In particular, Φ1 (𝐸) = 𝐸 if and only if 𝐸 is finite dimensional. Proof. (1) Φ1 (𝐸) = Φ2 (𝐸) follows from Theorem 3.13, since 𝐸 as a Banach lattice with order continuous norm is Dedekind complete. As already mentioned, each 𝑎 ∈ Γ𝐸 is a finite element of 𝐸, so span(Γ𝐸 ) ⊂ Φ1 (𝐸), since Φ1 (𝐸) is a vector sublattice¹⁰ (and an ideal) of 𝐸. For 𝜑 ∈ 𝐸 is a finite element, Theorem 3.11 implies that the principal band {𝜑}⊥⊥ has an order unit. Since {𝜑}⊥⊥ is a Banach lattice with respect to its order unit norm, by Proposition 1.2.13, Corollary 1.2.14 of [95] and the Kakutani–Bohnenblust– Kreins Theorem (Theorem 2.15), the band {𝜑}⊥⊥ is lattice isomorphic to an 𝐴𝑀-space 𝐶(𝐾) for some compact Hausdorff space 𝐾, and therefore the lattice operations in {𝜑}⊥⊥ are weakly sequentially continuous. Since the norm in the Banach lattice {𝜑}⊥⊥ is order continuous, Corollary 2.3 of [39] implies that {𝜑}⊥⊥ is atomic. From the previous proposition we have {𝜑}⊥⊥ = span(𝐻), where 𝐻 = Γ𝐸 ∩ {𝜑}⊥⊥ . The closed unit ball

10 If Γ𝐸 = 0, then we define span(Γ𝐸 ) = {0}.

32 | 3 Finite, totally finite and selfmajorizing elements of {𝜑}⊥⊥ , as a subset of an order interval, is compact (Theorem 2.8), therefore {𝜑}⊥⊥ is finite dimensional and 𝐻 is a finite set, {𝜑}⊥⊥ = span(𝐻) ⊂ span(Γ𝐸 ), which means that Φ1 (𝐸) ⊂ span(Γ𝐸 ). (2) Based on the first part of the theorem it is clear that Φ1 (𝐸) is not closed in 𝐸 if Γ𝐸 is infinite. For some classical normed vector lattices¹¹, one immediately obtains the following information on their finite elements: (a) if 𝐸 is one of the vector lattices c0 or ℓ𝑝 with 1 ≤ 𝑝 < ∞, then Φ1 (𝐸) = Φ2 (𝐸) = span(Γ𝐸 ) = span{𝑒𝑘 : 𝑘 = 1, 2, . . .}, where 𝑒𝑘 ∈ 𝐸 is the sequence which 𝑘-th entry equals, and all others are 0; (b) if 𝐸 = 𝐿 𝑝 (𝑎, 𝑏) for 1 ≤ 𝑝 < ∞ is the vector lattice of all classes of power-𝑝 integrable functions 𝑓 on the real interval [𝑎, 𝑏], then Φ1 (𝐸) = {0}. As mentioned after the definition of a totally finite element, the vector lattice K(ℝ) of all finite continuous functions on ℝ is a simple example of a vector lattice 𝐸 possessing the property Φ1 (𝐸) = Φ2 (𝐸) = 𝐸. The next theorem contains a characterization of the class of Banach lattices having this property . Theorem 3.19 (Characterization of Banach lattices with Φ1 (𝐸) = Φ2 (𝐸) = 𝐸). For a Banach lattice 𝐸, the following statements are equivalent: (1) Φ1 (𝐸) = Φ2 (𝐸) = 𝐸. (2) 𝐸 is lattice isomorphic to an 𝐴𝑀-space and each principal band has a generalized order unit. Proof. (2) ⇒ (1) follows straightforwardly from Theorem 3.15. (1) ⇒ (2). If Φ1 (𝐸) = 𝐸 then, due to Theorem 3.9, it suffices to show that 𝐸 is lattice isomorphic to an 𝐴𝑀-space. According to Theorem 2.1.12 of [95], it suffices to show that 1 (𝑥𝑛 )𝑛∈ℕ is order bounded in 𝐸 whenever 𝑥𝑛 ∈ 𝐸+ with 𝑥𝑛 → 0. Put 𝑢 = ∑∞ 𝑛=1 2𝑛 𝑥𝑛 , then ⊥⊥ 𝑥𝑛 ∈ {𝑢} for each 𝑛. Since 𝑢 is finite it follows from Theorem 3.15 and Remark 3.16 (1), that the sequence (𝑥𝑛) is order bounded in 𝐸 as (𝑥𝑛) is norm bounded in {𝑢}⊥⊥ . The following example shows that 𝐸 may fail to have an order unit even if 𝐸 is a Dedekind complete 𝐴𝑀-space and Φ1 (𝐸) = Φ2 (𝐸) = 𝐸. Example 3.20. A Dedekind complete 𝐴𝑀-space with Φ1 (𝐸) = Φ2 (𝐸) = 𝐸 without order unit, where each principal band has an order unit. Let 𝐽 be an uncountable index set, and 𝐸 the space of all bounded real functions 𝑓 on 𝐽, such that 𝑓(𝑗) = 0 for all but countable many 𝑗 ∈ 𝐽. Under the pointwise defined algebraic operations, the pointwise order, and equipped with the supremum norm, it is easy

11 For the order continuity of the norms in the vector lattices c0 and ℓp with 1 ≤ 𝑝 < ∞ see e. g., [8, Theorems 13.8, 10.7], and for 𝐿 𝑝 (𝑎, 𝑏) see p. 13 and [95, Theorem 2.4.2].

3.3 Finite elements in sublattices and in direct sums of Banach lattices

33

|

to verify that 𝐸 is a Dedekind complete 𝐴𝑀-space without order unit. But each principal band obviously has an order unit, so it follows that Φ1 (𝐸) = Φ2 (𝐸) = 𝐸.

3.3 Finite elements in sublattices and in direct sums of Banach lattices If 𝐸 is a vector lattice and 𝐻 a vector sublattice of 𝐸, then it is of interest to study the relations between Φ1 (𝐸) and Φ1 (𝐻), i. e., to ask whether or under which conditions the following relations hold: (i) Φ1 (𝐻) ⊂ Φ1 (𝐸),

(ii) Φ1 (𝐸) ∩ 𝐻 ⊂ Φ1 (𝐻),

(iii) Φ1 (𝐸) ∩ 𝐻 = Φ1 (𝐻).

(3.6)

We will show that the answers to these questions in general are negative, even if 𝐻 is supposed to be a more qualified sublattice, e. g., an order ideal or a band. Some sufficient conditions and counterexamples are also provided (see [37]). In what follows, among others, we shall consider three natural situations where a given vector lattice 𝐸 is embedded in another vector lattice. Let 𝐸 be normed vector lattice, then denote its norm completion and its bidual by 𝐸 and 𝐸󸀠󸀠 respectively. For an Archimedean vector lattice denote its Dedekind completion by 𝐸𝛿 . In each case, 𝐸 is naturally embedded as a vector sublattice into the ambient vector lattice 𝐸, 𝐸󸀠󸀠 , 𝐸𝛿 , respectively. In Subsection 3.3.1 we deal with 𝐸 and 𝐸𝛿 , and in 3.3.2 with 𝐸󸀠󸀠 . Finite elements in direct sums of Banach lattices are considered in Subsection 3.3.3.

3.3.1 Finite elements in sublattices Let 𝐸 be a normed vector lattice. Then its norm completion 𝐸 is a Banach lattice, and 𝐸 is a norm-dense vector sublattice in 𝐸. In general, the inclusion Φ1 (𝐸) ⊂ Φ1 (𝐸) does not hold. The two extreme cases in this situation are as follows: (1) If the vector lattice c00 is equipped with the supremum norm, then it is not norm complete. The norm completion of c00 is the Banach lattice c0. In this case Φ1 (c00 ) = c00 = Φ1 (c0 ). (2) The vector lattice 𝐶[0, 1], equipped with the integral-norm induced from 𝐿 1 (0, 1), is not norm complete but 𝐿 1 (0, 1) is its norm completion. One has Φ1 (𝐶[0, 1]) = 𝐶[0, 1] and Φ1 (𝐿 1 (0, 1)) = {0}. Before dealing with the Dedekind completion, we prove a general result about majorizing sublattices. Let 𝐸 be an Archimedean vector lattice and 𝐻 ⊂ 𝐸 a vector sublattice.

34 | 3 Finite, totally finite and selfmajorizing elements Theorem 3.21. If the vector sublattice 𝐻 majorizes the vector lattice 𝐸, i. e., ∀ 𝑥 ∈ 𝐸,

∃ 𝑦 ∈ 𝐻 with 𝑥 ≤ 𝑦,

(3.7)

then 𝜑 ∈ Φ1 (𝐻) if and only if 𝜑 ∈ Φ1 (𝐸) ∩ 𝐻, i. e., Φ1 (𝐻) = Φ1 (𝐸) ∩ 𝐻. Proof. If 𝜑 ∈ Φ1 (𝐻) then |𝑦| ∧ 𝑛|𝜑| ≤ 𝑐𝑥 𝑧 for all 𝑦 ∈ 𝐻 and 𝑛 ∈ ℕ, where 𝑧 > 0 is an 𝐻-majorant of 𝜑. For any 𝑥 ∈ 𝐸 take 𝑦 ∈ 𝐻 such that |𝑥| ≤ 𝑦. Then |𝑥| ∧ 𝑛|𝜑| ≤ 𝑦 ∧ 𝑛|𝜑| ≤ 𝑐𝑦 𝑧 . It follows that 𝜑 ∈ Φ1 (𝐸), and 𝑧 is even an 𝐸-majorant of 𝜑. If 𝜑 ∈ 𝐻 and 𝜑 ∈ Φ1 (𝐸) with the 𝐸-majorant 𝑧, then according to (3.7), take 𝑦 ∈ 𝐻 such that |𝑧| ≤ 𝑦. It is easy to verify that 𝑦 is an 𝐻-majorant of 𝜑, so 𝜑 ∈ Φ1 (𝐻). If 𝐸𝛿 denotes the Dedekind completion of an Archimedean vector lattice 𝐸, then Φ1 (𝐸) = Φ1 (𝐸𝛿 ) ∩ 𝐸, i. e., relation (iii) of (3.6) is true. This follows from the preceding theorem, since 𝐸 can be identified with a vector sublattice 𝐻 in 𝐸𝛿 possessing the property (3.7) (see Formula 2.1 on p. 7). Thus we have Corollary 3.22. It holds that 𝜑 ∈ Φ1 (𝐸) if and only if 𝜑 ∈ Φ1 (𝐸𝛿 ) ∩ 𝐸, i. e., Φ1 (𝐸) = Φ1 (𝐸𝛿 ) ∩ 𝐸. Concerning the inclusion (i) in (3.6), we mention two extreme cases. (1) Assume that the inclusion (i) is always true whenever 𝐻 is an arbitrary ideal of a Banach lattice 𝐸. Then Φ1 (𝐸) = 𝐸, i. e., every element 𝑥 of the Banach lattice 𝐸 must be finite. This follows from the fact that the principal ideal 𝐼𝑥 generated by 𝑥 in 𝐸 contains |𝑥| as an order unit, and therefore satisfies Φ1 (𝐼𝑥 ) = 𝐼𝑥 (see Theorem 3.6). So, by assumption, 𝑥 ∈ 𝐼𝑥 = Φ1 (𝐼𝑥 ) ⊂ Φ1 (𝐸) holds for any 𝑥 ∈ 𝐸. It is clear that Φ1 (𝐼𝑥 ) ⊂ Φ1 (𝐸) for each 𝑥 ∈ 𝐸 suffices to conclude that Φ1 (𝐸) = 𝐸. (2) The relations Φ1 (𝐻) ≠ {0} = Φ1 (𝐸) are also possible as the following example shows. Example 3.23. Let 𝐸 = 𝐿 𝑝 [0, 1], with 1 ≤ 𝑝 < ∞, and 𝑥𝑛 ∈ 𝐸+ (𝑛 ∈ ℕ) pairwise disjoint 𝑝 elements such that ‖𝑥𝑛‖ = 1. Let 𝐻 = {∑∞ 𝑛=1 𝜆 𝑛 𝑥𝑛 : (𝜆 𝑛 ) ∈ ℓ }. Then it can be verified that (a) 𝐻 is a norm closed vector sublattice (but not an ideal) of 𝐸, which is lattice isomorphic to ℓ𝑝 ; (b) there is a positive contractive projection 𝑃 from 𝐸 onto 𝐻 (see [95, Theorem 2.7.11]); (c) Φ1 (𝐻) = span{𝑥𝑛}, but Φ1 (𝐸) = {0} (see Theorem 3.18). We draw the following conclusions: The inclusion (i) may be false in the cases where (1) 𝐻 is an arbitrary order ideal of 𝐸; (2) 𝐻 is a norm closed sublattice which is the range of a positive projection on 𝐸. However, if 𝐻 is a closed ideal of a Banach lattice we have the following theorem.

3.3 Finite elements in sublattices and in direct sums of Banach lattices

| 35

Theorem 3.24. Let 𝐸 be a Banach lattice and 𝐻 a closed order ideal of 𝐸. Then Φ1 (𝐻) ⊂ Φ1 (𝐸). In particular, if 𝐻 is a band of 𝐸, then Φ1 (𝐻) ⊂ Φ1 (𝐸). Proof. If the element 𝜑 ∈ 𝐻 is finite in 𝐻, then Theorem 3.15 (here we use the closedness of 𝐻 in order to guarantee that 𝐻 is a Banach lattice) implies that 𝑧 ∈ 𝐻+ = 𝐻∩𝐸+ ⊥⊥ exists such that 𝐵({𝜑}⊥⊥ 𝐻 ) ⊂ [−𝑧, 𝑧]𝐻 . Now for any 𝑦 ∈ 𝐵({𝜑}𝐸 ) by means of Formula (2.2) the relation |𝑦| = sup{|𝑦| ∧ 𝑛|𝜑|} holds. Since 𝐻 is an ideal, the elements |𝜑| and |𝑦|∧𝑛|𝜑| belong to 𝐻. Hence, |𝑦|∧𝑛|𝜑| ∈ 𝐵({𝜑}⊥⊥ 𝐻 ) as ‖ |𝑦| ∧ 𝑛|𝜑| ‖ ≤ ‖𝑦‖ ≤ 1. It follows that |𝑦| ∧ 𝑛|𝜑| ≤ 𝑧

for all 𝑛 ∈ ℕ.

Thus |𝑦| ≤ 𝑧, i. e., 𝐵({𝜑}⊥⊥ 𝐸 ) ⊂ [−𝑧, 𝑧]. Again, Theorem 3.15 yields that 𝜑 is finite in 𝐸. We now discuss the inclusion (ii) in (3.6) and start with two examples. Example 3.25. Let 𝐸 = ℓ∞ and 𝐻 = c0. Then 𝐻 is a closed ideal of 𝐸, Φ1 (𝐸) = 𝐸, and Φ1 (𝐻) = c00 = span{𝑒𝑘 : 𝑘 = 1, 2, . . .}, where 𝑒𝑘 denotes the element of 𝐻 with 𝑘’s entry equal to 1, and all others are 0. Therefore, Φ1 (𝐸) ∩ 𝐻 = 𝐻 ⊄ Φ1 (𝐻). Another example shows that even for a band 𝐻 inclusion (ii) may fail. Example 3.26. Let 𝐸 = 𝐶[0, 2], and 𝐻𝑐 = {𝑥 ∈ 𝐸 : 𝑥(𝑡) = 0, ∀𝑡 ∈ (0, 𝑐)} for arbitrary 𝑐 ∈ (0, 2). Then (a) Φ1 (𝐸) = 𝐸; (b) 𝐻𝑐 is a band of 𝐸 and Φ1 (𝐻𝑐 ) = ⋃𝑢>𝑐 𝐻𝑢 , so that Φ1 (𝐸) ∩ 𝐻𝑐 ⊄ Φ1 (𝐻𝑐 ) for all 𝑐 ∈ (0, 2). Proof. (a) is obvious, as 𝐸 has an order unit. For (b) it is clear that 𝐻𝑐 is an ideal for each 𝑐 ∈ (0, 2). 𝐻𝑐 is even a band¹² in 𝐸, since (0, 𝑐) is a regularly open subset of [0, 2]. Now we show that Φ1 (𝐻𝑐 ) = ⋃𝑢>𝑐 𝐻𝑢 , i. e., an element 𝜑 ∈ 𝐻𝑐 is finite (and, of course, automatically totally finite) if there is a positive number 𝛿𝜑 such that 𝜑(𝑡) = 0 for 𝑡 ∈ (0, 𝑐 + 𝛿𝜑 ). For 𝑢 > 𝑐 it is easy to see that the function 𝑥𝜆,𝑢 ∈ 𝐸, for each 𝜆 ∈ (𝑐, 𝑢), defined by 0, 𝑡 ∈ [0, 𝜆] { { 𝑥𝜆,𝑢 (𝑡) = { linear, 𝑡 ∈ (𝜆, 𝑢] { 1, 𝑡 ∈ (𝑢, 2] , { belongs to 𝐻𝑐 , and is a generalized order unit of the band 𝐻𝑢 for each 𝜆 ∈ (𝑐, 𝑢). Since any band in a Banach lattice is closed (see [95, Proposition 1.2.3]), the band 𝐻𝑐 is a

12 In general, if 𝑄 is a topological space and 𝐴 ⊂ 𝑄 an arbitrary regularly open subset of 𝑄, i. e., int(𝑐𝑙(𝐴)) = 𝐴, then 𝐻𝐴 = {𝑥 ∈ 𝐶(𝑄) : 𝑥(𝑡) = 0 for 𝑡 ∈ 𝐴} is a band in 𝐶(𝑄); see [144, Example 9.4].

36 | 3 Finite, totally finite and selfmajorizing elements Banach lattice in its own right. Observe that 𝐻𝑢 is a principal band (cf. Remark 3.16 (2)) and so, Theorems 3.9 and 3.15 yield 𝐻𝑢 ⊂ Φ1 (𝐻𝑐 ). On the other hand, if 𝜑 ∈ 𝐻𝑐 is a finite element in 𝐻𝑐 , let 𝑡∗ = inf{𝑡 ∈ [𝑐, 2] : |𝜑(𝑡)| > 0}. Then we claim that 𝑡∗ > 𝑐. Otherwise, the sequence 𝑡𝑘 > 𝑐, 𝑡𝑘 → 𝑐 exists such that |𝜑(𝑡𝑘 )| > 0 for all 𝑘 ∈ ℕ. Since 𝜑 is finite in 𝐻𝑐 , by definition, 0 ≤ 𝑧 ∈ 𝐻𝑐 exists such that for each 𝑥 ∈ 𝐻𝑐 there is a real 𝑐𝑥 > 0 with |𝑥| ∧ 𝑛|𝜑| ≤ 𝑐𝑥 𝑧 for

𝑛∈ℕ.

It is easy to see that 𝑧(𝑡𝑘) > 0 for 𝑘 ∈ ℕ. By taking 𝑧0 = √𝑧 ∈ 𝐻𝑐 we obtain for some 𝑐𝑧0 𝑧0 ∧ 𝑛|𝜑| ≤ 𝑐𝑧0 𝑧

for 𝑛 ∈ ℕ .

It follows that 𝑧0 (𝑡𝑘 ) ∧ 𝑛|𝜑(𝑡𝑘 )| ≤ 𝑐𝑧0 𝑧(𝑡𝑘)

for 𝑛, 𝑘 ∈ ℕ .

This implies 1 ≤ 𝑐𝑧0 √𝑧(𝑡𝑘 ), which is impossible as 𝑧(𝑡𝑘 ) → 𝑧(𝑐) = 0. Therefore, 𝑡∗ > 𝑐, i. e., 𝜑 ∈ 𝐻𝑡∗ so that b) holds. We now draw the corresponding conclusions: The inclusion (ii) may be false in the cases (1) 𝐻 is a closed ideal of 𝐸; (2) 𝐻 is a band of 𝐸. If the sublattice 𝐻 is the range of a positive projection on 𝐸, then the situation is much better. Theorem 3.27. Let 𝐸 be a vector lattice and 𝐻 a sublattice of 𝐸. If there is a positive projection 𝑃 from 𝐸 onto 𝐻, then Φ1 (𝐸) ∩ 𝐻 ⊂ Φ1 (𝐻). Proof. If 𝜑 ∈ Φ1 (𝐸) and its 𝐸-majorant 𝑧 ∈ 𝐸+ , then for each 𝑥 ∈ 𝐸 there is |𝑥| ∧ 𝑛|𝜑| ≤ 𝑐𝑥 𝑧 for 𝑛 ∈ ℕ for some real 𝑐𝑥 > 0. It follows from the positivity of 𝑃 that 𝑃(|𝑥| ∧ 𝑛|𝜑|) ≤ 𝑐𝑥 𝑃𝑧 = 𝑐𝑥 𝑧0

for 𝑛 ∈ ℕ

where 𝑧0 = 𝑃𝑧 ∈ 𝐻. In particular, if now 𝜑 belongs to 𝐻 and if 𝑥 ∈ 𝐻 then |𝑥|∧𝑛|𝜑| ∈ 𝐻 and, hence |𝑥| ∧ 𝑛|𝜑| ≤ 𝑐𝑥 𝑧0 for 𝑛 ∈ ℕ , which shows that 𝜑 is finite in 𝐻, i. e., 𝜑 ∈ Φ1 (𝐻). For a projection band 𝐻 of 𝐸, the next result shows that (iii), and therefore also the relations (i) and (ii), hold. Theorem 3.28. Let 𝐻 be a projection band in a vector lattice 𝐸, and 𝑃𝐻 the band projection from 𝐸 onto 𝐻. Then 𝑃𝐻 (Φ1 (𝐸)) = Φ1 (𝐸) ∩ 𝐻 = Φ1 (𝐻).

3.3 Finite elements in sublattices and in direct sums of Banach lattices

|

37

Proof. For 𝜑 ∈ Φ1 (𝐸) and some 𝐸-majorant 𝑧 ∈ 𝐸, one has |𝑥|∧𝑛|𝜑| ≤ 𝑐𝑥 𝑧 for any 𝑥 ∈ 𝐸, some 𝑐𝑥 > 0, and all 𝑛 ∈ ℕ. Applying 𝑃𝐻 to this inequality yields 𝑃𝐻 (|𝑥| ∧ 𝑛|𝜑|) ≤ 𝑐𝑥𝑃𝐻 𝑧 = 𝑐𝑥 𝑧0

for 𝑛 ∈ ℕ ,

(3.8)

where 𝑧0 = 𝑃𝐻 𝑧 ∈ 𝐻. If now 𝑥 ∈ 𝐻, then |𝑥| ∧ 𝑛|𝜑| ∈ 𝐻 and 𝑃𝐻 (|𝑥| ∧ 𝑛|𝜑|) = |𝑥| ∧ 𝑛|𝜑|

for 𝑛 ∈ ℕ.

(3.9)

Using the representation of 𝜑 = 𝑃𝐻 𝜑 + 𝜑0 , where 𝜑0 ⊥ 𝑥 for any 𝑥 ∈ 𝐻 (see [144, Theorem 11.4]), and |𝑃𝐻 𝜑 + 𝜑0 | = |𝑃𝐻 𝜑| ∨ |𝜑0 |, one has |𝑥| ∧ 𝑛|𝜑| = |𝑥| ∧ 𝑛(|𝑃𝐻 𝜑| ∨ |𝜑0 |) = |𝑥| ∧ 𝑛|𝑃𝐻 𝜑| and so, according to (3.8) and (3.9) , |𝑥| ∧ 𝑛|𝑃𝐻 𝜑| ≤ 𝑐𝑥𝑧0 , i. e., 𝑃𝐻 𝜑 ∈ Φ1 (𝐻). That means 𝑃𝐻 (Φ1 (𝐸)) ⊂ Φ1 (𝐻). If 𝜓 ∈ Φ1 (𝐻), then there is an 𝐻-majorant 0 < ℎ0 ∈ 𝐻, such that for any 𝑥 ∈ 𝐻 the inequality |𝑥| ∧ 𝑛|𝜓| ≤ 𝑐𝑥 ℎ0 holds for some 𝑐𝑥 > 0, and all 𝑛 ∈ ℕ. If 𝑥 is an arbitrary element of 𝐸, then |𝑥| = 𝑃𝐻 |𝑥|+𝑥󸀠 , where 𝑥󸀠 ⊥ ℎ for any ℎ ∈ 𝐻, in particular, 𝑥󸀠 ∧|𝜓| = 0. This yields |𝑥| ∧ 𝑛|𝜓| = (𝑃𝐻 |𝑥| + 𝑥󸀠 ) ∧ 𝑛|𝜓| = 𝑃𝐻 |𝑥| ∧ 𝑛|𝜓| ≤ 𝑐𝑃𝐻 |𝑥| ℎ0

for 𝑛 ∈ ℕ .

Consequently 𝜓 ∈ Φ1 (𝐸), which shows that Φ1 (𝐻) ⊂ Φ1 (𝐸). Now one has 𝑃𝐻 (Φ1 (𝐸)) ⊂ Φ1 (𝐻) ⊂ Φ1 (𝐸) ∩ 𝐻 = 𝑃𝐻 (Φ1 (𝐸) ∩ 𝐻) ⊂ 𝑃𝐻 (Φ1 (𝐸)), which proves the required equations. Since in a Dedekind complete vector lattice every band is a projection band one has the following corollary. Corollary 3.29. Let 𝐻1 , . . . , 𝐻𝑛 be bands in a Dedekind complete vector lattice 𝐸. If 𝐸 = 𝐻1 ⊕ 𝐻2 ⊕ ⋅ ⋅ ⋅ ⊕ 𝐻𝑛 , then Φ1 (𝐸) = Φ1 (𝐻1 ) ⊕ Φ1 (𝐻2 ) ⊕ ⋅ ⋅ ⋅ ⊕ Φ1 (𝐻𝑛 ). Indeed, if 𝐸 = 𝐻1 ⊕⋅ ⋅ ⋅⊕𝐻𝑛 , and 𝑃𝑖 are the band projections onto 𝐻𝑖 , then by the theorem 𝑃𝑖Φ1 (𝐸) = Φ1 (𝐻𝑖 ), 𝑖 = 1, . . . , 𝑛. Therefore each 𝜑 ∈ Φ1 (𝐸) has a unique representation as 𝜑 = 𝜑1 + ⋅ ⋅ ⋅ + 𝜑𝑛 , where 𝜑𝑖 ∈ Φ1 (𝐻𝑖 ).

3.3.2 Finite elements in the bidual of Banach lattices In this section it will be shown that every finite element in a Banach lattice 𝐸 is also finite in its bidual 𝐸󸀠󸀠 . To do this we need the following proposition, which might be of independent interest. Proposition 3.30. Let 𝐸 be a Banach lattice and 𝐻 a closed order ideal of 𝐸. If 𝑖 : 𝐻 → 𝐸 is the inclusion mapping, and 𝑖 󸀠 : 𝐸󸀠 → 𝐻󸀠 is the adjoint mapping of 𝑖, then

38 | 3 Finite, totally finite and selfmajorizing elements (1) the kernel space 𝑁(𝑖 󸀠 ) = {𝑓 ∈ 𝐸󸀠 : 𝑖 󸀠 (𝑓) = 0} of 𝑖 󸀠 is a band in 𝐸󸀠 and 𝐸󸀠 = 𝑁(𝑖 󸀠 ) ⊕ 𝑁(𝑖 󸀠 )⊥ ; (2) the restriction 𝑖 󸀠 |𝑁(𝑖 󸀠 )⊥ of 𝑖 󸀠 to 𝑁(𝑖 󸀠 )⊥ is an isometric Riesz isomorphism from 𝑁(𝑖 󸀠 )⊥ onto 𝐻󸀠 . Proof. (1) It is clear that 𝑖 󸀠 (𝑓) = 𝑓|𝐻 , i. e., the value of 𝑖 󸀠 (𝑓) is the restriction of the functional 𝑓 to 𝐻 for each 𝑓 ∈ 𝐸󸀠 . So 𝑓 ∈ 𝑁(𝑖 󸀠 ) if and only if 𝑓(𝑥) = 0 for all 𝑥 ∈ 𝐻. Now if 𝑓𝛼 ∈ 𝑁(𝑖 󸀠 ) is a net such that 𝑓𝛼 ↑ 𝑓 in 𝐸󸀠 , then 𝑓(𝑥) = sup𝑓𝛼 (𝑥) for all 𝑥 ∈ 𝐸+ (see [95, Corollary 1.3.4]). In particular, 𝑓(𝑥) = sup𝑓𝛼 (𝑥) = 0 for all 𝑥 ∈ 𝐻+ , and hence for all 𝑥 ∈ 𝐻, i. e., 𝑓 ∈ 𝑁(𝑖 󸀠 ), and 𝑁(𝑖 󸀠 ) is a band in 𝐸󸀠 . Obviously 𝐸󸀠 = 𝑁(𝑖 󸀠 ) ⊕ 𝑁(𝑖 󸀠 )⊥ , as 𝐸󸀠 is Dedekind complete and moreover, 𝑖 󸀠 |𝑁(𝑖 󸀠 )⊥ is injective. (2) Since 𝑖 is (trivially) an interval-preserving lattice homomorphism, then by Theorem 1.4.19 of [95], 𝑖 󸀠 and hence 𝑖 󸀠 |𝑁(𝑖 󸀠 )⊥ are also interval-preserving lattice homomorphisms. Therefore it suffices to show that ‖𝑖 󸀠 (𝑓)‖ = ‖𝑓‖ for each 𝑓 ∈ 𝑁(𝑖 󸀠 )⊥ . Let 𝑔 ∈ 𝐸󸀠 be an extension of 𝑓|𝐻 such that ‖𝑔‖ = ‖𝑓|𝐻 ‖ (the existence is guaranteed by the Hahn– Banach Theorem), then 𝑔 − 𝑓 ∈ 𝑁(𝑖 󸀠 ), and |𝑔| = |𝑔 − 𝑓| + |𝑓|. It follows that ‖𝑖 󸀠 (𝑓)‖ = ‖𝑓|𝐻 ‖ ≤ ‖𝑓‖ ≤ ‖𝑔‖ = ‖𝑓|𝐻 ‖ = ‖𝑖 󸀠 (𝑓)‖ as desired. Theorem 3.31. Let 𝐸 be a Banach lattice and 𝑗 : 𝐸 → 𝐸󸀠󸀠 the canonical embedding. Then 𝑗(Φ1 (𝐸)) ⊂ Φ1 (𝐸󸀠󸀠 ). Proof. For an arbitrary element 𝜑 ∈ 𝐸, the set 𝐻 = {𝜑}⊥⊥ is a closed ideal in 𝐸, as in a normed vector lattice every band is closed. If 𝑖 : 𝐻 → 𝐸 denotes the corresponding inclusion mapping, then by Proposition 3.30 we have 𝐸󸀠 = 𝑁(𝑖 󸀠 ) ⊕ 𝑁(𝑖 󸀠 )⊥ , and hence 󸀠

󸀠

𝐸󸀠󸀠 = (𝑁(𝑖 󸀠 )) ⊕ (𝑁(𝑖 󸀠 )⊥ ) . It is easy to see that (𝑗𝜑)(𝑓) = 𝑓(𝜑) = 0 for all 𝑓 ∈ 𝑁(𝑖 󸀠 ), therefore the functional 󸀠 𝑗𝜑 must belong to (𝑁(𝑖 󸀠 )⊥ ) . Proposition 3.30 then yields 𝐻󸀠 ≅ 𝑁(𝑖 󸀠 )⊥ (isometric Riesz 󸀠 isomorphic), so that 𝐻󸀠󸀠 ≅ (𝑁(𝑖 󸀠 )⊥ ) . If 𝜑 is now a finite element of 𝐸, then 𝐻 possesses a generalized order unit, say 𝑧. The order ideal 𝐻 = {𝜑}⊥⊥ , equipped with the norm ‖𝑥‖𝑧 = inf{𝜆 > 0 : |𝑥| ≤ 𝜆𝑧}

for each

𝑥∈𝐻

makes 𝐻 lattice isomorphic to an 𝐴𝑀-space (see the proof of Theorem 3.15). It follows that 𝐻󸀠󸀠 is lattice isomorphic to an 𝐴𝑀-space with an order unit. Theorem 3.6 and Theorem 3.27 imply Φ1 (𝐻󸀠󸀠 ) = 𝐻󸀠󸀠 , and hence by Theorem 3.28 󸀠

󸀠

(𝑁(𝑖 󸀠 )⊥ ) = Φ1 ((𝑁(𝑖 󸀠 )⊥ )󸀠 ) = Φ1 (𝐸󸀠󸀠 ) ∩ (𝑁(𝑖 󸀠 )⊥ ) . 󸀠

As 𝑗𝜑 belongs to (𝑁(𝑖󸀠 )⊥ ) , it is clear that 𝑗𝜑 is a finite element in 𝐸󸀠󸀠 .

3.3 Finite elements in sublattices and in direct sums of Banach lattices

|

39

Remark 3.32. If 𝐸 is identified with 𝑗(𝐸) in 𝐸󸀠󸀠 , then Φ1 (𝐸) ⊂ Φ1 (𝐸󸀠󸀠 ) ∩ 𝐸. The inverse inclusion is false in general. Take 𝐸 = c0 . Then 𝐸󸀠󸀠 = ℓ∞ and compare with Example 3.25.

3.3.3 Finite elements in direct sums of Banach lattices For an arbitrary set 𝐼, denote by F(𝐼) the set of all finite subsets of 𝐼 and order it by inclusion, i. e., for j, k ∈ F(𝐼) we write j ≤ k if j ⊆ k. Let (𝜉𝑖 )𝑖∈𝐼 be a family of real numbers. Then its finite sums 𝜎j = ∑ 𝜉𝑖 ,

j ∈ F(𝐼)

𝑖∈j

compose a net. A family of numbers (𝜉𝑖)𝑖∈𝐼 is called summable if the net (𝜎j )j∈F(𝐼) converges. Denote the limit of (𝜎j )j∈F(𝐼) by 𝜎. Then it is called the sum of the family (𝜉𝑖)𝑖∈𝐼 , which is written as 𝜎 = ∑𝐼 𝜉𝑖 . Let 𝐼 be an arbitrary set and 𝑋𝑖 be a Banach lattice for each 𝑖 ∈ 𝐼, where the norm in each space 𝑋𝑖 is denoted by ‖ ⋅ ‖𝑖 . Consider the following spaces: 𝐸0 = c0(𝐼, 𝑋𝑖 ) = {(𝑥𝑖)𝑖∈𝐼 : 𝑥𝑖 ∈ 𝑋𝑖 , ∀𝜀 > 0 ∃ finite set 𝐼𝜀 ⊂ 𝐼 with ‖𝑥𝑖 ‖𝑖 < 𝜀, ∀𝑖 ∉ 𝐼𝜀 }; 𝐸𝑝 = ℓ𝑝 (𝐼, 𝑋𝑖 ) = {(𝑥𝑖 )𝑖∈𝐼 : 𝑥𝑖 ∈ 𝑋𝑖 , ∑𝑖∈𝐼 ‖𝑥𝑖 ‖𝑖 < ∞} for 𝑝

∞

𝐸

𝑝 ∈ ℕ;

∞

= ℓ (𝐼, 𝑋𝑖 ) = {(𝑥𝑖 )𝑖∈𝐼 : 𝑥𝑖 ∈ 𝑋𝑖 , sup𝑖∈𝐼 ‖𝑥𝑖 ‖𝑖 < ∞}.

Under the pointwise defined linear operations and order, i. e., for 𝜆, 𝜇 ∈ ℝ 𝜆(𝑥𝑖 )𝑖∈𝐼 + 𝜇(𝑦𝑖 )𝑖∈𝐼 = (𝜆𝑥𝑖 + 𝜇𝑦𝑖 )𝑖∈𝐼

for (𝑥𝑖 )𝑖∈𝐼 , (𝑦𝑖 )𝑖∈𝐼 ∈ 𝐸0 , 𝐸𝑝 , 𝐸∞ ,

and for all 𝑖 ∈ 𝐼 ,

(𝑥𝑖 )𝑖∈𝐼 ≤ (𝑦𝑖)𝑖∈𝐼 ⇔ 𝑥𝑖 ≤ 𝑦𝑖 and the norms defined by sup ‖𝑥𝑖 ‖𝑖 , { { 𝑖∈𝐼 { 1 ‖𝑥‖ = ‖(𝑥𝑖 )𝑖∈𝐼 ‖ = { 𝑝 { { (∑ ‖𝑥 ‖𝑝 ) , 𝑖 𝑖 { 𝑖∈𝐼

if 𝑥 ∈ 𝐸0 or 𝑥 ∈ 𝐸∞ if 𝑥 ∈ 𝐸𝑝 ,

, 𝑝∈ℕ

respectively, the spaces 𝐸0 , 𝐸∞ and 𝐸𝑝 , 𝑝 ∈ ℕ are Banach lattices and are called direct sums (for details see [33]). Let 𝐽𝑗 : 𝑋𝑗 → 𝐸𝑝 denote the canonical lattice embeddings into the spaces 𝐸𝑝 , for 𝑝 = 0, 𝑝 = ∞ and 𝑝 ∈ ℕ, i. e., 𝐽𝑗 𝑥 = (𝑥𝑖 )𝑖∈𝐼 = {

0, 𝑥,

𝑖 ≠ 𝑗 𝑖=𝑗

for 𝑥 ∈ 𝑋𝑗 .

40 | 3 Finite, totally finite and selfmajorizing elements Then 𝐽𝑗 𝑋𝑗 is a projection band in 𝐸𝑝 for 𝑝 = 0, 𝑝 = ∞, and 𝑝 ∈ ℕ, and if 𝑃𝑗 : 𝐸𝑝 → 𝐽𝑗 𝑋𝑗 denotes the band projection from 𝐸𝑝 onto 𝐽𝑗 𝑋𝑗 , where 𝑃𝑗((𝑥𝑖 )𝑖∈𝐼 ) = 𝐽𝑗 𝑥𝑗 , then by Theorem 3.28 𝐽𝑗 Φ1 (𝑋𝑗 ) = Φ1 (𝐽𝑗 𝑋𝑗 ) ⊂ Φ1 (𝐸𝑝 ) . (3.10) The finite elements in the Banach lattices 𝐸𝑝 are characterized as follows. Theorem 3.33. With the notations from above the following statements hold: (1) For the spaces 𝐸0 and 𝐸𝑝 with 𝑝 ∈ ℕ : (𝜑𝑖 )𝑖∈𝐼 ∈ Φ1 (𝐸𝑝 ) if and only if 𝜑𝑖 ∈ Φ1 (𝑋𝑖 ) for all 𝑖 ∈ 𝐼, and 𝜑𝑖 = 0 for all but finite many 𝑖 ∈ 𝐼. (2) For the space 𝐸∞ : (𝜑𝑖 )𝑖∈𝐼 ∈ Φ1 (𝐸∞ ) if and only if 𝜑𝑖 ∈ Φ1 (𝑋𝑖 ) for all 𝑖 ∈ 𝐼, and 0 ≤ 𝑧𝑖 ∈ 𝑋𝑖 exists such that 𝐵{𝜑𝑖 }⊥⊥ ⊂ [−𝑧𝑖 , 𝑧𝑖 ], and sup𝑖∈𝐼 ‖𝑧𝑖 ‖𝑖 < ∞. Proof. (1) The sufficiency is clear from (3.10), since the considered family (𝜑𝑖 )𝑖∈𝐼 , has only a finite number of nonzero coordinates 𝜑𝑖 . Then, by definition, 𝐽𝑖 𝜑𝑖 is the family in 𝐸𝑝 having 𝜑𝑖 as its 𝑖-th coordinate and 0 elsewhere. Due to the linearity of the spaces Φ1 (𝐸𝑝 ) for 𝑝 = 0, and 𝑝 ∈ ℕ, the element (𝜑𝑖 )𝑖∈𝐼 belongs to Φ1 (𝐸𝑝 ), for 𝑝 = 0 and 𝑝 ∈ ℕ respectively. For necessity the argument is as follows: if 𝜑 = (𝜑𝑖 )𝑖∈𝐼 ∈ Φ1 (𝐸𝑝 ), (𝑝 = 0, 𝑝 ∈ ℕ), then Theorem 3.28, applied to our situation, says 𝑃𝑖 Φ1 (𝐸𝑝 ) = Φ1 (𝐸𝑝 ) ∩ 𝐽𝑖 𝑋𝑖 = Φ1 (𝐽𝑖 𝑋𝑖 ) , and therefore yields 𝑃𝑖 𝜑 = 𝐽𝑖 𝜑𝑖 ∈ Φ1 (𝐽𝑖 𝑋𝑖 ). According to (3.10) we have 𝜑𝑖 ∈ Φ1 (𝑋𝑖 ) for all 𝑖 ∈ 𝐼. Now we claim that 𝜑𝑖 = 0 for all but finite many 𝑖 ∈ 𝐼. Assume there would be a sequence (𝑖𝑘 )𝑘∈ℕ ⊂ 𝐼 such that ‖𝜑𝑖𝑘 ‖ > 0 for 𝑘 ∈ ℕ. Put 𝜓𝑖𝑘 = ⊥⊥

|𝜑𝑖 | 𝑘

‖𝜑𝑖 ‖ 𝑘

. Then

𝐽𝑖𝑘 𝜓𝑖𝑘 ∈ 𝐵{𝜑}⊥⊥ = 𝐵𝐸𝑝 ∩ {𝜑} for all 𝑘 ∈ ℕ. According to Theorem 3.15 an element 𝑧 = (𝑧𝑖)𝑖i𝐼 ∈ 𝐸𝑝 exists such that 0 ≤ 𝑧 and 𝐵{𝜑}⊥⊥ ⊂ [−𝑧, 𝑧], from where 𝜓𝑖𝑘 ≤ 𝑧𝑖𝑘 follows, and hence 1 = ‖𝜓𝑖𝑘 ‖ ≤ ‖𝑧𝑖𝑘 ‖ for 𝑘 ∈ ℕ is easily obtained. This is impossible, as 𝑧 ∈ 𝐸𝑝 . Thus (1) holds. (2) The necessity is clear from the proof above. For sufficiency we mention only the fact that 𝐵{𝜑}⊥⊥ = {(𝑥𝑖)𝑖∈𝐼 ∈ 𝐸∞ : 𝑥𝑖 ∈ 𝐵{𝜑𝑖 }⊥⊥ , ∀𝑖 ∈ 𝐼}. The next example shows that the conditions 𝜑𝑖 ∈ Φ1 (𝑋𝑖 ) and ‖𝜑𝑖 ‖𝑖 ≤ 1 for all 𝑖 ∈ 𝐼 are not sufficient for an element 𝜑 = (𝜑𝑖 )𝑖∈𝐼 to be finite in 𝐸∞ . This means the element 𝑧 = (𝑧𝑖 )𝑖∈𝐼 consisting of the 𝑧𝑖 ∈ 𝑋𝑖 mentioned in the second statement of the theorem must belong to 𝐸∞ . Consequently, the condition sup𝑖∈𝐼 ‖𝑧𝑖 ‖𝑖 < ∞ in the theorem cannot be dropped. Example 3.34. Let 𝑋𝑛 = (c, ‖ ⋅ ‖𝑛 ) with the norm ‖(𝜆 𝑘 )‖𝑛 = max{‖(𝜆 𝑘 )‖, 𝑛 lim |𝜆 𝑘 |}, 𝑘→∞

where ‖(𝜆 𝑘 )‖ = sup𝑘 |𝜆 𝑘 | is the usual norm on c.

(𝜆 𝑘)𝑘∈ℕ ∈ c,

3.4 Selfmajorizing elements in vector lattices

|

41

Put 𝐸 = ℓ∞ (ℕ, 𝑋𝑛 ) = ℓ∞ (𝑋𝑛). Then (a) 𝑋𝑛 is an 𝐴𝑀-space with an order unit, and hence Φ1 (𝑋𝑛) = 𝑋𝑛 for 𝑛 ∈ ℕ; (b) 𝐸 is an 𝐴𝑀-space and the element 𝜑 = (𝜑𝑖 )𝑖∈𝐼 exists in 𝐸 such that 𝜑𝑖 ∈ Φ1 (𝑋𝑖 ), and ‖𝜑𝑖 ‖ ≤ 1 for all 𝑖 ∈ 𝐼, but 𝜑 is not finite in 𝐸. Proof. Assertion (a) and 𝐸 being an 𝐴𝑀-space are clear. For the remaining part of (b) take 𝜑 = (𝜑𝑛 )𝑛∈ℕ ∈ 𝐸 with 𝜑𝑛 = (1, 12 , . . . , 𝑘1 , . . .) ∈ 𝑋𝑛 . Then 𝜑𝑛 ∈ Φ1 (𝑋𝑛 ) = 𝑋𝑛 (by Theorem 3.6), and ‖𝜑𝑛 ‖𝑛 = 1 for all 𝑛 ∈ ℕ. It is easy to verify that 𝑋𝑛 = {𝜑𝑛 }⊥⊥ for each 𝑛 ∈ ℕ, and hence 𝐸 = {𝜑}⊥⊥ . For the element 𝜑 not to be finite, by Theorem 3.15, it suffices to show that the closed unit ball 𝐵𝐸 of 𝐸 is not order bounded. Indeed, if there would be an 𝑧 = (𝑧𝑛)𝑛∈ℕ ∈ 𝐸+ with 𝐵𝐸 ⊂ [−𝑧, 𝑧], then 𝐵𝑋𝑛 ⊂ [−𝑧𝑛, 𝑧𝑛 ] for the closed unit balls in 𝑋𝑛 , 𝑛 ∈ ℕ. Let 𝑒𝑘 be the element in 𝑋𝑛 with 𝑘’th entry equal to 1 and all others are 0, then 𝑒1 + ⋅ ⋅ ⋅ + 𝑒𝑘 ∈ 𝐵𝑋𝑛 and, therefore, 𝑒1 + ⋅ ⋅ ⋅ + 𝑒𝑘 ≤ 𝑧𝑛 for all 𝑘 ∈ ℕ. It follows that 𝑒 = (1, 1, ⋅ ⋅ ⋅ , 1, ⋅ ⋅ ⋅ ) ≤ 𝑧𝑛, and ‖𝑧𝑛‖𝑛 ≥ ‖𝑒‖𝑛 = 𝑛 for each 𝑛 ∈ ℕ, which is impossible since 𝑧 = (𝑧𝑛)𝑛∈ℕ ∈ 𝐸.

3.4 Selfmajorizing elements in vector lattices In this section we study a special class of finite elements accentuated by having the modulus among their majorants. Definition 3.35. An element 𝜑 of an Archimedean vector lattice 𝐸 is called selfmajorizing if |𝜑| is a majorant of 𝜑, i. e., for each element 𝑥 ∈ 𝐸 there is a constant 𝑐𝑥 > 0 such that there holds the inequality: |𝑥| ∧ 𝑛|𝜑| ≤ 𝑐𝑥|𝜑|

for all 𝑛 ∈ ℕ.

(3.11)

Selfmajorizing elements in Archimedean vector lattices appeared under the name semi-order units in [85], later in [46] and at times in [38]. A characterization of them had already been given by W. A. Luxemburg an L. C. Moore Jr. in [85] (see Theorem 3.42 below). Semi-order unit elements were thoroughly investigated in a different context in the paper of W. A. Feldman and J. F. Porter [46]. For each semi-order unit a natural seminorm is defined in the vector lattice, and it is shown that under some conditions the corresponding locally convex topology is metrizable, such that due to the order continuity of the norm, the vector lattice can be represented as a dense subspace of 𝐶(𝑄) for some locally compact topological space 𝑄. These authors also proved some duality results between semi-order unit spaces and the so-called semibase spaces. In order to expose the close relation of semi-order units to finite elements we will use here the notion of selfmajorizing element instead of semi-order unit. Within the scope of studying finite and totally finite elements in Archimedean vector lattices, the selfmajorizing elements occasionally occurred, e. g., in [36] and [54]. The present section is devoted to a systematic study of this class of finite elements, see [114].

42 | 3 Finite, totally finite and selfmajorizing elements Selfmajorizing elements in vector lattices of operators are dealt with in Chapter 4, where more examples of selfmajorizing elements are also provided.

3.4.1 The order ideal of all selfmajorizing elements in a vector lattice The set of all selfmajorizing elements in 𝐸 is denoted by 𝑆(𝐸) , the set of positive selfmajorizing elements by 𝑆+(𝐸), i. e., 𝑆+ (𝐸) = 𝑆(𝐸) ∩ 𝐸+ . It is clear that together with 𝜑 󵄨 󵄨 also 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 is a selfmajorizing element. First we deal with the set of positive selfmajorizing elements, then introduce the order ideal of the selfmajorizing elements. Proposition 3.36. The set 𝑆+ (𝐸) has the following properties: (1) the set 𝑆+ (𝐸) is a cone, i. e., 𝑥, 𝑦 ∈ 𝑆+ (𝐸), and 𝜆 ≥ 0 imply 𝑥 + 𝑦, 𝜆𝑥 ∈ 𝑆+ (𝐸), and −𝑥, 𝑥 ∈ 𝑆+ (𝐸) imply 𝑥 = 0; (2) if 𝑥, 𝑦 ∈ 𝑆+ (𝐸), then also 𝑥 ∨ 𝑦, 𝑥 ∧ 𝑦 ∈ 𝑆+ (𝐸). Proof. (1) Let 𝑥, 𝑦 ∈ 𝑆+(𝐸). For arbitrary 𝑣 ∈ 𝐸 there are constants 𝑐𝑣, 𝑑𝑣 > 0, such that |𝑣| ∧ 𝑛𝑥 ≤ 𝑐𝑣 𝑥 and |𝑣| ∧ 𝑛𝑦 ≤ 𝑑𝑣 𝑦 for all 𝑛 ∈ ℕ. So we get |𝑣| ∧ 𝑛(𝑥 + 𝑦) ≤ (|𝑣| ∧ 𝑛𝑥) + (|𝑣| ∧ 𝑛𝑦) ≤ 𝑐𝑣𝑥 + 𝑑𝑣 𝑦 ≤ max {𝑐𝑣 , 𝑑𝑣 } (𝑥 + 𝑦) for all 𝑛 ∈ ℕ, which shows that 𝑥 + 𝑦 is a selfmajorizing element. The fact that any positive multiple of a selfmajorizing element is again in 𝑆+ (𝐸) follows immediately from the definition. Due to 𝑆+ (𝐸) ⊆ 𝐸+ , one has −𝑥, 𝑥 ∈ 𝑆+ (𝐸) imply 𝑥 = 0. Therefore 𝑆+ (𝐸) is a cone. (2) Let the elements and constants be as in the first part of the proof. Then by applying the distributive law in vector lattices we get |𝑣| ∧ 𝑛(𝑥 ∨ 𝑦) = (|𝑣| ∧ 𝑛𝑥) ∨ (|𝑣| ∧ 𝑛𝑦) ≤ 𝑐𝑣 𝑥 ∨ 𝑑𝑣 𝑦 ≤ max {𝑐𝑣 , 𝑑𝑣 } (𝑥 ∨ 𝑦), and so 𝑥 ∨ 𝑦 ∈ 𝑆+ (𝐸). For the element 𝑥 ∧ 𝑦 |𝑣| ∧ 𝑛(𝑥 ∧ 𝑦) = (|𝑣| ∧ 𝑛𝑥) ∧ (|𝑣| ∧ 𝑛𝑦) ≤ 𝑐𝑣𝑥 ∧ 𝑑𝑣 𝑦 ≤ max {𝑐𝑣 , 𝑑𝑣 } (𝑥 ∧ 𝑦) implies 𝑥 ∧ 𝑦 ∈ 𝑆+ (𝐸). In general, 𝑥 ∈ 𝐸+ , 𝑧 ∈ 𝑆+ (𝐸) and 𝑥 ≤ 𝑧 do not imply 𝑥 ∈ 𝑆+ (𝐸), as the forthcoming Example (3.47) will show. Proposition 3.37. Let 𝜑 be an arbitrary element in a vector lattice 𝐸. The following statements are equivalent: (1) 𝜑 is a selfmajorizing element; (2) |𝜑| is in 𝑆+ (𝐸); (3) the elements 𝜑+ and 𝜑− belong to 𝑆+ (𝐸).

3.4 Selfmajorizing elements in vector lattices

|

43

Proof. According to the definition, the equivalence of (1) and (2) is clear. (2) ⇒ (3). Let 𝑥 be an arbitrary element of 𝐸. By assumption, there is a number 𝑐𝑥 > 0 such that |𝑥| ∧ 𝑛(𝜑+ + 𝜑− ) ≤ 𝑐𝑥 (𝜑+ + 𝜑− ) for all 𝑛 ∈ ℕ. It follows that |𝑥| ∧ 𝑛(𝜑+ + 𝜑− ) ∧ 𝑛𝜑+ ≤ 𝑐𝑥 (𝜑+ + 𝜑− ) ∧ 𝑛𝜑+ for any 𝑛 ∈ ℕ. Because of 𝑛(𝜑+ + 𝜑− ) ≥ 𝑛𝜑+ , the left-hand side of the last inequality is equal to |𝑥| ∧ 𝑛𝜑+ . For the right-hand side one has + + − + 𝑐𝑥 (𝜑+ + 𝜑− ) ∧ 𝑛𝜑+ ≤ (𝑐⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑥 𝜑 ∧ 𝑛𝜑 ) + (𝑐 𝑥 𝜑 ∧ 𝑛𝜑 ). ≤𝑐𝑥 𝜑+ +

=0

+

Therefore, |𝑥| ∧ 𝑛𝜑 ≤ 𝑐𝑥 𝜑 for all 𝑛 ∈ ℕ. This shows that 𝜑+ is in 𝑆+(𝐸). Analogously, 𝜑− ∈ 𝑆+ (𝐸) can be shown. (3) ⇒ (2). Since 𝑆+ (𝐸) is a cone, 𝜑+ ∈ 𝑆+ (𝐸) and 𝜑− ∈ 𝑆+ (𝐸) imply 𝜑+ + 𝜑− = |𝜑| ∈ 𝑆+(𝐸). Corollary 3.38. The set 𝑆(𝐸) of all selfmajorizing elements has the following properties: (1) if 𝑥 ∈ 𝑆(𝐸), then 𝜆𝑥 ∈ 𝑆(𝐸) for any 𝜆 ∈ ℝ; (2) if 𝑥, 𝑦 ∈ 𝑆(𝐸) are disjoint, then 𝑥 + 𝑦 ∈ 𝑆(𝐸); (3) for 𝑥 and 𝑦, 𝑥 ∨ 𝑦 and 𝑥 ∧ 𝑦 ∈ 𝑆(𝐸) hold. 󵄨 󵄨 Proof. Due to |𝜆𝑥| = |𝜆||𝑥|, the assertion (1) is clear. Assertion (2) follows from |𝑥|∨󵄨󵄨󵄨𝑦󵄨󵄨󵄨 = 󵄨󵄨 󵄨 󵄨󵄨𝑥 + 𝑦󵄨󵄨󵄨 for 𝑥 ⊥ 𝑦. We prove (3). Let 𝑥, 𝑦 ∈ 𝑆(𝐸). Due to the equality 𝑥∧𝑦 = −((−𝑥)∨(−𝑦)), it suffices to show 𝑥 ∨ 𝑦 ∈ 𝑆(𝐸). We use |𝑥 ∨ 𝑦| = (𝑥 ∨ 𝑦)+ + (𝑥 ∨ 𝑦)− = (𝑥+ ∨ 𝑦+ ) + (𝑥− ∧ 𝑦− ). In view of Proposition 3.37, the elements 𝑥+ , 𝑥− , 𝑦+ , 𝑦− are in 𝑆+(𝐸), and so by Proposition 3.36 we have |𝑥 ∨ 𝑦| ∈ 𝑆+ (𝐸). Again by Proposition 3.37, we obtain 𝑥 ∨ 𝑦 ∈ 𝑆(𝐸). The set ̃ 𝑆(𝐸) = {𝜑 ∈ 𝐸 : ∀𝑥 ∈ 𝐸 ∃ 𝑐𝑥 > 0 such that |𝑥| ∧ 𝑛|𝜑| ≤ 𝑐𝑥|𝜑| for ∀𝑛 ∈ ℕ}, which would be the natural candidate to be considered in this context, fails to be linear as the following example demonstrates. In the vector lattice ℓ∞ of all bounded sequences¹³ , the elements 𝜑1 = (2, 32 , 43 , 54 , ...) and 𝜑2 = (−1, −1, −1, −1, ...) both belong ̃ to 𝑆(𝐸). However, the element 𝜑1 + 𝜑2 = (1, 12 , 13 , 14 , ...) does not. Therefore the set Φ3 (𝐸) = 𝑆+ (𝐸) − 𝑆+ (𝐸) is the candidate of interest, which turns out to be an order ideal and appropriate for comparison with the ideals Φ1 (𝐸) and Φ2 (𝐸). The latter will be done systematically in Section 6.2.

13 This vector lattice is considered in detail in Example 3.47 in Section 3.4.3.

44 | 3 Finite, totally finite and selfmajorizing elements Proposition 3.39. The set Φ3 (𝐸) is an order ideal in 𝐸. Proof. Since 𝑆+ (𝐸) is a cone, the set Φ3 (𝐸) is a linear subspace of 𝐸. It remains to show 󵄨 󵄨 that Φ3 (𝐸) is solid, i. e., that 𝑥 ∈ 𝐸, 𝜑 ∈ Φ3 (𝐸) and |𝑥| ≤ 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 imply 𝑥 ∈ Φ3 (𝐸). First, let 𝑥 be an element of 𝐸+ , and 𝜑 ∈ 𝑆+ (𝐸) with 𝑥 ≤ 𝜑. Since 𝜑 is selfmajorizing for each 𝑦 ∈ 𝐸, there is a constant 𝑐𝑦 such that the inequalities 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨𝑦󵄨󵄨 ∧ 𝑛(𝑥 + 𝜑) ≤ 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ∧ 2𝑛𝜑 ≤ 𝑐𝑦 𝜑 ≤ 𝑐𝑦 (𝑥 + 𝜑) hold for all 𝑛 ∈ ℕ. This shows that 𝑥 + 𝜑 ∈ 𝑆+ (𝐸), and so 𝑥 = (𝑥 + 𝜑) − 𝜑 ∈ Φ3 (𝐸). Now let 𝑥 be an arbitrary element of 𝐸, and 𝜑 = 𝜑1 − 𝜑2 an element of Φ3 (𝐸), where 𝜑1 , 𝜑2 ∈ 𝑆+ (𝐸). Assume |𝑥| ≤ |𝜑|. Then 𝑥+ + 𝑥− = |𝑥| ≤ |𝜑| = 𝜑+ + 𝜑− ≤ 𝜑1 + 𝜑2 , and therefore 𝑥+ ≤ 𝜑1 + 𝜑2 , and 𝑥− ≤ 𝜑1 + 𝜑2 . The element 𝜑1 + 𝜑2 is in 𝑆+ (𝐸), and the elements 𝑥+ and 𝑥− are positive. From the first part of the proof it follows that 𝑥+ , 𝑥− ∈ Φ3 (𝐸). The element 𝑥 = 𝑥+ − 𝑥− belongs to Φ3 (𝐸) due to the linearity of the latter. This shows that Φ3 (𝐸) is solid. Corollary 3.40. The ideal Φ3 (𝐸) coincides with the ideal generated by the set of all selfmajorizing elements 𝑆(𝐸). Proof. Clearly 𝑆+ (𝐸) ⊆ 𝑆(𝐸) implies Φ3 (𝐸) ⊆ 𝐼𝑆(𝐸) , and by Proposition 3.37 one has 𝑆(𝐸) ⊆ Φ3 (𝐸). Therefore, 𝐼𝑆(𝐸) ⊆ 𝐼Φ3 (𝐸) = Φ3 (𝐸). The set Φ3 (𝐸) is called the ideal of all selfmajorizing elements. The inclusion 𝑆+(𝐸) ⊆ Φ3 (𝐸) ∩ 𝐸+ is obvious. In general, the two sets do not coincide, as Example 3.47 shows. Since 𝑆+ (𝐸) ⊆ Φ2 (𝐸) implies Φ3 (𝐸) = 𝐼𝑆+ (𝐸) ⊆ 𝐼Φ2 (𝐸) = Φ2 (𝐸), the ideal Φ3 (𝐸) is always contained in the ideal Φ2 (𝐸). Example 3.49 will show that Φ3 (𝐸) ≠ Φ2 (𝐸) in general. Remark 3.41. At the beginning of the proof of the previous proposition it is shown that 𝑥 + 𝜑 ∈ 𝑆+ (𝐸), whenever 𝑥 ∈ 𝐸, 𝜑 ∈ 𝑆+ (𝐸) and 𝑥 ≤ 𝜑.

3.4.2 General properties of selfmajorizing elements In Proposition 3.8 it was proved that any atom in 𝐸 is a selfmajorizing element¹⁴ of 𝐸. In Theorem 4.2 it will be shown that a surjective lattice isomorphism between Dedekind complete Banach lattices is a selfmajorizing element in the vector lattice of regular

14 Indeed, put 𝑐𝑥 = 𝜆𝑎 (|𝑥|) there, then |𝑥| ∧ 𝑛 𝑎 ≤ 𝑐𝑥 𝑎.

3.4 Selfmajorizing elements in vector lattices

| 45

operators. Similar to the other kinds of finite elements, a topological characterization of the selfmajorizing elements will be found in Chapter 6. In the current section we provide some general properties of selfmajorizing elements. The main characterization of selfmajorizing elements is contained in the subsequent theorem, which goes back to [85] and [46]. For completeness we provide its proof. Theorem 3.42. For an element 𝜑 of a vector lattice 𝐸, the following statements are equivalent: (1) the element 𝜑 is selfmajorizing; (2) the principal ideal 𝐼𝜑 generated by 𝜑 is a projection band in 𝐸. Proof. (1) ⇒ (2). Let 𝜑 be a selfmajorizing element. By assumption, for each 𝑥 ∈ 𝐸+ there is a constant 𝑐𝑥 such that 𝑥 ∧ 𝑛|𝜑| ≤ 𝑐𝑥 |𝜑| for all 𝑛 ∈ ℕ. Hence, 𝑥 ∧ 𝑛|𝜑| ≤ 𝑥 ∧ 𝑐𝑥 |𝜑| also holds for all 𝑛 ≥ 𝑐𝑥 . Therefore the set {𝑥 ∧ 𝑛|𝜑| : 𝑛 ∈ ℕ} contains only a finite number of different elements, and so there exists sup {𝑥 ∧ 𝑛|𝜑| : 𝑛 ∈ ℕ} ≤ 𝑥 ∧ 𝑐𝑥 |𝜑|. (3.12) ⊥⊥

Theorem 3.13 of [9] implies that {𝜑} is a projection band. It remains to show that ⊥⊥ ⊥⊥ {𝜑} = 𝐼𝜑 . The inclusion 𝐼𝜑 ⊂ {𝜑}⊥⊥ is clear. For {𝜑} ⊂ 𝐼𝜑 let 𝑦 be an arbitrary ⊥⊥ ⊥⊥ element in {𝜑} . Then also |𝑦| ∈ {𝜑} and, again by the same theorem, one has |𝑦| = 𝑃𝜑 |𝑦| = sup {|𝑦| ∧ 𝑛|𝜑| : 𝑛 ∈ ℕ} . The inequality (3.12) implies |𝑦| ≤ |𝑦| ∧ 𝑐𝑦 |𝜑| ≤ 𝑐𝑦 |𝜑| and so 𝑦 ∈ 𝐼𝜑 . (2) ⇒ (1). Let 𝜑 be an element of 𝐸 such that 𝐼𝜑 is a projection band. Then 𝐼𝜑 = {𝜑}⊥⊥ and the projection 𝑃𝜑 : 𝐸 → 𝐼𝜑 of the element 𝑦 ∈ 𝐸+ is given by 𝑃𝜑 𝑦 = sup {𝑦 ∧ 𝑛|𝜑| : 𝑛 ∈ ℕ} . For an arbitrary 𝑥 ∈ 𝐸, the element 𝑃𝜑 |𝑥| is in 𝐼𝜑 , where |𝜑| is an order unit. Therefore 𝑃𝜑 |𝑥| ≤ 𝑐𝑥 |𝜑| for some 𝑐𝑥 > 0. Hence we get |𝑥| ∧ 𝑛|𝜑| ≤ 𝑃𝜑 |𝑥| ≤ 𝑐𝑥 |𝜑| for all 𝑛 ∈ ℕ. So, 𝜑 is a selfmajorizing element in 𝐸. Remark 3.43. In the first part of the proof we established that for each selfmajorizing element 𝜑 ∈ 𝐸, its band {𝜑}⊥⊥ is a projection band, and that 𝐼𝜑 = {𝜑}⊥⊥ . Similar to Theorem 3.6 we have Proposition 3.44. Let 𝐸 be a vector lattice. Then (1) any order unit in a vector lattice is a positive selfmajorizing element; (2) if the vector lattice 𝐸 possesses an order unit then Φ3 (𝐸) = Φ2 (𝐸) = Φ1 (𝐸) = 𝐸.

46 | 3 Finite, totally finite and selfmajorizing elements Proof. (1) If 𝑢 is an order unit in 𝐸, then 𝐸 = 𝐼𝑢 . The result¹⁵ follows immediately from the previous Theorem 3.42, since 𝐸 is obviously a projection band. (2) According to the first part, the order unit 𝑢 is a selfmajorizing element, and so 𝐸 = 𝐼𝑢 ⊆ Φ3 (𝐸), i. e., Φ3 (𝐸) = 𝐸. If 𝐸 possesses an order unit 𝑢, then the proposition tells us that any 𝑥 ∈ 𝐸 has a representation 𝑥 = 𝑥1 − 𝑥2 , where 𝑥1 , 𝑥2 ∈ 𝑆+ (𝐸). How to find such 𝑥1 and 𝑥2 ? If, for an arbitrary element 𝑥 ∈ 𝐸, one has |𝑥| ≤ 𝜆 𝑥 𝑢, then 𝑥+ ≤ 𝜆 𝑥 𝑢 and 𝑥− ≤ 𝜆 𝑥 𝑢. Since 𝑢 ∈ 𝑆+(𝐸), and due to Remark 3.41, one has 𝑥+ +𝜆 𝑥 𝑢, 𝑥− +𝜆 𝑥 𝑢 ∈ 𝑆+ (𝐸). A required representation for 𝑥 is now 𝑥 = (𝑥+ + 𝜆 𝑥 𝑢) − (𝑥− + 𝜆 𝑥 𝑢). A characterization of selfmajorizing elements in a vector lattice with the principal projection property is given next; cf. the characterization of finite elements in Theorem 3.11. Theorem 3.45. Let 𝐸 be a vector lattice with the principal projection property (𝑝𝑝𝑝). For an element 𝜑 ∈ 𝐸, the following statements are equivalent: (1) the element 𝜑 is selfmajorizing; (2) the principal ideal 𝐼𝜑 coincides with the principal band {𝜑}⊥⊥ ; 󵄨 󵄨 (3) the element 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 is an order unit in {𝜑}⊥⊥ . 󵄨 󵄨 Proof. Since 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 is an order unit in 𝐼|𝜑| , and 𝐼𝜑 = 𝐼|𝜑| , then (2) ⇒ (3). We prove (3) ⇒ 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 (2). If 󵄨󵄨𝜑󵄨󵄨 is an order unit in {𝜑}⊥⊥ , then 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ≤ 𝜆 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 for some 𝜆 > 0, which shows that ⊥⊥ ⊥⊥ 𝑦 ∈ 𝐼𝜑 for any 𝑦 ∈ {𝜑} . In view of 𝐼𝜑 ⊆ {𝜑} , we get 𝐼𝜑 = {𝜑}⊥⊥ . The equivalence (1) ⇔ (2) follows from Theorem 3.42, since {𝜑}⊥⊥ by assumption is a projection band, and 𝐼𝜑 = {𝜑}⊥⊥ . The principal projection property (𝑝𝑝𝑝) of 𝐸 in the last theorem was used only in order to conclude that {𝜑}⊥⊥ is a projection band, and to establish the implication (2) ⇒ (1). If the vector lattice 𝐸 does not possess the principal projection property but is supposed to be a Banach lattice instead, then, similar to the situation we faced with the characterization of finite elements at the beginning of Section 3.2, the same characterization of selfmajorizing elements can be proved, where the proof only slightly differs from that of Theorem 3.15. Theorem 3.46. Let 𝐸 be a Banach lattice. Then for an element 𝜑 ∈ 𝐸 the statements (1), (2), and (3) of Theorem 3.45 are equivalent. Proof. (1) ⇒ (2) follows from Theorem 3.42. The proof of the equivalence of (2) and (3) is the same as in the previous theorem. Therefore it remains to show (3) ⇒ (1). For ⊥⊥ the proof we exploit the idea of the proof of Theorem 3.15. The band {𝜑} is (norm) ⊥⊥ closed in 𝐸, and therefore ({𝜑} , ‖ ⋅ ‖) is a Banach space. According to the assumption

15 This result can be directly derived from the definition without referring to Theorem 3.42.

3.4 Selfmajorizing elements in vector lattices ⊥⊥

on {𝜑}

|

47

, the unit order norm ‖𝑥‖𝜑 = inf {𝜆 > 0; |𝑥| < 𝜆|𝜑|} , ⊥⊥

makes the normed space ({𝜑}

⊥⊥

𝑥 ∈ {𝜑}

, ‖ ⋅ ‖𝜑 ) into an 𝐴𝑀-space and 󵄨 󵄨 |𝑥| ≤ ‖𝑥‖𝜑 󵄨󵄨󵄨𝜑󵄨󵄨󵄨

(3.13)

holds. Then ‖𝑥‖ ≤ ‖𝑥‖𝜑 ‖𝜑‖, and by the open mapping theorem both norms ‖⋅‖ and ‖⋅‖𝜑 ⊥⊥ ⊥⊥ are equivalent on {𝜑} . In particular, ‖𝑥‖𝜑 ≤ 𝐶‖𝑥‖ for some 𝐶 > 0, and all 𝑥 ∈ {𝜑} . ⊥⊥ 󵄨 󵄨 The inequality (3.13) implies now |𝑥| ≤ 𝐶‖𝑥‖ 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 for each 𝑥 ∈ {𝜑} . ⊥⊥ Let 𝑥 be an arbitrary element of 𝐸. Since |𝑥| ∧ 𝑛|𝜑| belongs to {𝜑} we have |𝑥| ∧ 󵄨󵄨 󵄨󵄨 𝑛|𝜑| ≤ 𝐶‖|𝑥| ∧ 𝑛|𝜑|‖ 󵄨󵄨𝜑󵄨󵄨 , and since |𝑥| ∧ 𝑛|𝜑| ≤ |𝑥| implies ‖|𝑥| ∧ 𝑛|𝜑|‖ ≤ ‖𝑥‖, we are able to estimate the right term and get |𝑥| ∧ 𝑛|𝜑| ≤ 𝐶‖𝑥‖|𝜑|, showing that the element 𝜑 is selfmajorizing.

3.4.3 Examples of selfmajorizing elements We inspect some examples of vector lattices and give a description of all, or at least some, selfmajorizing elements in them. Observe that later in Section 6.1.4, when a topological characterization of selfmajorizing elements is available, we are able to describe all selfmajorizing elements in the vector lattice c by means of this characterization; see Example 6.28. Example 3.47. Let 𝐸 be the vector lattice ℓ∞ of all real bounded sequences. It is well known (see Proposition 3.44) that 𝐸 = Φ1 (𝐸) = Φ2 (𝐸) = Φ3 (𝐸). The selfmajorizing elements in 𝐸 are precisely the sequences (𝑥𝑛)𝑛∈ℕ , for which inf {|𝑥𝑛 | : 𝑥𝑛 ≠ 0} > 0.

(3.14)

We show first that each sequence which satisfies the property (3.14) is a selfmajorizing element. Let 𝑥 be such a sequence, where its corresponding infimum in (3.14) is denoted by 𝜀. Let 𝑦 = (𝑦𝑛 )𝑛∈ℕ ∈ ℓ∞ be an arbitrary element, and 𝑐𝑦󸀠 > 0 a constant such that |𝑦𝑛 | ≤ 𝑐𝑦󸀠 for all 𝑛 ∈ ℕ. If 𝑛 ∈ ℕ is an index with 𝑥𝑛 = 0, then |𝑦𝑛 | ∧ 𝑚|𝑥𝑛 | ≤ 𝑐𝑦󸀠 |𝑥𝑛 |, since for each 𝑚 ∈ ℕ both sides are zero. If 𝑛 ∈ ℕ is an index with |𝑥𝑛 | > 0, then |𝑦𝑛 | ∧ 𝑚|𝑥𝑛 | ≤ |𝑦𝑛 | ≤ 𝑐𝑦󸀠 ≤ 𝑐𝑦󸀠 1𝜀 |𝑥𝑛 | for each 𝑚 ∈ ℕ , and so |𝑦| ∧ 𝑚|𝑥| ≤ 𝑐𝑦 |𝑥| for all 𝑚 ∈ ℕ, where 𝑐𝑦 = 1𝜀 𝑐𝑦󸀠 . This shows that 𝑥 ∈ 𝑆(𝐸). Let now 𝑥 ∈ 𝑆(𝐸) and assume inf {|𝑥𝑛 | : 𝑥𝑛 ≠ 0} = 0. Then there is a subsequence (𝑛𝑘 )𝑘∈ℕ , such that |𝑥𝑛𝑘 | > 0 and |𝑥𝑛𝑘 | → 0. For the element 1 = (1, 1, 1, ..) and an appropriate constant 𝐶 > 0 we get that for each 𝑘 ∈ ℕ 1 ∧ 𝑚|𝑥𝑛𝑘 | ≤ 𝐶|𝑥𝑛𝑘 | holds for all 𝑚 ∈ ℕ.

48 | 3 Finite, totally finite and selfmajorizing elements If 𝑚 (for fixed 𝑘) chosen is sufficiently large, then 1 = 1 ∧ 𝑚|𝑥𝑛𝑘 | ≤ 𝐶|𝑥𝑛𝑘 | or, equivalently, |𝑥𝑛𝑘 | ≥ 𝐶1 , which contradicts |𝑥𝑛𝑘 | → 0. As a consequence, the element 𝜑 = (1, 12 , 13 , 14 , ...) ∈ 𝐸+ is not selfmajorizing in 𝐸. Nevertheless, we know from Proposition 3.44 that Φ3 (𝐸) = 𝐸, so it is clear that 𝑆+ (𝐸) ≠ Φ3 (𝐸) ∩ 𝐸+ , since 𝜑 ∈ Φ3 (𝐸) ∩ 𝐸+ . Moreover, for the element 𝜑, the inequality 𝜑 ≤ 1 does not imply 𝜑 ∈ 𝑆+ (𝐸), although 1 ∈ 𝑆+ (𝐸). Due to Remark 3.41, one has 𝜑 + 1 ∈ 𝑆+ (𝐸), and so for 𝜑 ∈ Φ3 (𝐸) the representation 𝜑 = (𝜑 + 1) − 1. For the finite and totally finite elements of 𝐸 we refer to Corollary 3.7. Observe that 𝐸 contains selfmajorizing elements which are not order units, e. g., the element (1, 0, 1, 0, . . .). Example 3.48. Let 𝐸 be the vector lattice s of all real sequences. The finite elements in 𝐸 are exactly the sequences with only a finite number of nonzero coordinates, i. e., Φ1 (s) = c00 . Indeed, if for an element 𝜑 = (𝜑1 , 𝜑2 , . . . , 𝜑𝑘 , . . .) ∈ Φ1 (𝐸) with a majorant 𝑧 = (𝑧1 , 𝑧2 , . . . , 𝑧𝑘 , . . .) ∈ 𝐸, one has 𝜑𝑘 ≠ 0 for infinite many indices 𝑘, 󵄨 󵄨 then for the element 𝑤 = (𝑤𝑘 )𝑘∈ℕ ∈ 𝐸 with 𝑤𝑘 = 𝑘 𝑧𝑘 , the inequality 𝑤 ∧ 𝑛 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 ≤ 𝑐𝑤 𝑧 yields 󵄨 󵄨 𝑘 𝑧𝑘 ∧ 𝑛 󵄨󵄨󵄨𝜑𝑘 󵄨󵄨󵄨 ≤ 𝑐𝑤 𝑧𝑘 for each 𝑘 ∈ ℕ. 󵄨 󵄨 If 𝜑𝑘 ≠ 0 and 𝑛 is sufficiently large, then 𝑧𝑘 > 0, and 𝑘𝑧𝑘 ∧ 𝑛 󵄨󵄨󵄨𝜑𝑘 󵄨󵄨󵄨 = 𝑘 𝑧𝑘 , which contradicts the last estimation if 𝑘 > 𝑐𝑤 . Since s is Dedekind complete, by Theorem 3.13 one has Φ2 (𝐸) = Φ1 (𝐸). It is easy to see that each finite element is also selfmajorizing. Therefore, in this vector lattice 𝑆(𝐸) = Φ3 (𝐸) = Φ2 (𝐸) = Φ1 (𝐸), and 𝑆+ (𝐸) = Φ3 (𝐸) ∩ 𝐸+ = Φ1 (𝐸) ∩ 𝐸+ . Example 3.49. Let 𝐸 be the vector lattice 𝐶(ℝ) of all real continuous functions on ℝ. As already mentioned, Φ1 (𝐸) coincides with Φ2 (𝐸), and is the vector sublattice K(ℝ) of all continuous functions with compact support. We show 𝑆+ (𝐸) = {0}, and so 𝑆(𝐸) = Φ3 (ℝ) = {0}. Assume there is an element 𝜑 with 0 < 𝜑 ∈ 𝑆+ (𝐸). Since the support of the function 𝜑 is compact, we define 𝑡0 = sup {𝑡 ∈ ℝ : 𝜑(𝑡) > 0}. Then 𝜑(𝑡0 ) = 0, and for any 𝑛 ∈ ℕ there is a point 𝑡𝑛 ∈ (𝑡0 − 𝑛1 , 𝑡0 ) with 𝜑(𝑡𝑛) > 0. Since 𝜑 ∈ Φ1 (𝐸) for the constant 1 function 1, a number 𝑐1 > 0 exists such that 1 ∧ 𝑚𝜑 ≤ 𝑐1𝜑

for all 𝑚 ∈ ℕ.

In particular, 1 ∧ 𝑚𝜑(𝑡𝑛) ≤ 𝑐1 𝜑(𝑡𝑛) for all natural numbers 𝑚 and 𝑛. If 𝑚 is chosen sufficiently large, then 𝑚𝜑(𝑡𝑛 ) > 1, and so 1 ≤ 𝑐1𝜑(𝑡𝑛), i. e., 𝜑(𝑡𝑛) ≥ 𝑐1 for all 𝑛 ∈ ℕ. This 1 is in contradiction to 𝜑 (𝑡𝑛) → 𝜑 (𝑡0 ) = 0. Example 3.50. Let 𝐸 be the vector lattice 𝐶 [0, 1] of all real continuous functions on the interval [0, 1]. Then 𝐸 = Φ𝑖 (𝐸) for 𝑖 = 1, 2, 3 according to Proposition 3.44. It is easy to see that the elements of 𝑆(𝐸) are characterized as follows: 𝜑 ∈ 𝑆(𝐸) ⇐⇒ min{|𝜑(𝑡)| : 𝑡 ∈ [0, 1]} > 0 ⇐⇒ |𝜑| is an order unit of 𝐶 [0, 1] .

3.5 Finite elements in ℓ-algebras and in product algebras

|

49

Example 3.51. Let 𝐸 be the vector lattice 𝐵 [0, 1] of all real bounded functions on the interval [0, 1]. The elements of 𝑆(𝐸) are characterized as follows: 𝜑 ∈ 𝑆(𝐸) ⇐⇒ inf {|𝜑(𝑡)| : 𝜑(𝑡) ≠ 0} > 0. The Dirichlet-function 𝜒|ℚ∩[0,1] is a selfmajorizing element in 𝐵 [0, 1]. Hence, in contrast to the previous example, the last vector lattice contains positive selfmajorizing elements which are not order units in 𝐸.

3.5 Finite elements in ℓ-algebras and in product algebras In this section, the finite elements are investigated in several classes of lattice-ordered algebras, which are known as ℓ-algebras. After the introduction of the special types of ℓ-algebras in Subsection 3.5.1, i. e., 𝑑-, 𝑓- and almost 𝑓-algebras, which are needed for our purposes, in Subsection 3.5.2 we consider finite elements in Archimedean ℓ-algebras with multiplicative unit, in 𝑓-algebras without multiplicative unit in Subsection 3.5.3, and finally in Subsection 3.5.4 in product algebras. The additional structure of an associative multiplication among the elements and appropriate requirements for compatibility of the vector lattice and algebra structures lead to some new questions and new properties concerning the collections of finite, totally finite and selfmajorizing elements; see [93]. In many cases, the order ideal of finite elements is an ring ideal as well. It is well known, and by far the most important example, that the vector lattice of all orthomorphisms on an Archimedean vector lattice is an Archimedean 𝑓-algebra with a weak order unit, see e. g., [9, Theorem 8.24]. This fact will be used several times in the current section. For details concerning ℓ-algebras we refer to the monographs [2, 9, 25, 95, 144], as well as to the papers [21, 22, 26, 27, 55, 103]. Recent developments in the theory of ℓ-algebras is reflected in the survey papers [28] and [29]. In this section we reproduce the results obtained in [93].

3.5.1 Lattice ordered algebras Further on, an algebra is understood to be a set A equipped with several operations: besides the addition (+), and the usual scalar multiplication which turn A into a vector space, there an associative multiplication (⋅) is also defined, satisfying the distributive laws 𝑎 ⋅ (𝑏 + 𝑐) = 𝑎 ⋅ 𝑏 + 𝑎 ⋅ 𝑐

and (𝑏 + 𝑐) ⋅ 𝑎 = 𝑏 ⋅ 𝑎 + 𝑐 ⋅ 𝑎

for all 𝑎, 𝑏, 𝑐 ∈ A.

50 | 3 Finite, totally finite and selfmajorizing elements –

A vector lattice A is called a lattice-ordered algebra, a Riesz algebra or also an ℓalgebra, if A is equipped with an associative multiplication¹⁶ such that A becomes an algebra, where (ℓ)

𝑎, 𝑏 ≥ 0 󳨐⇒ 𝑎𝑏 ≥ 0 holds for all 0 ≤ 𝑎, 𝑏 ∈ A.

The basic notions and properties of ℓ-algebras can be found in [143, Chapter 20]. Equivalent to (ℓ) are the conditions: (ℓ1 ) if 𝑎, 𝑏, 𝑐 ∈ A satisfy 𝑎 ≤ 𝑏, and 𝑐 ≥ 0, then 𝑎𝑐 ≤ 𝑏𝑐,

–

(ℓ2 ) |𝑎𝑏| ≤ |𝑎||𝑏| for all 𝑎, 𝑏 ∈ A; see [21, Sect. 1]. An ℓ-algebra is called a 𝑑-algebra (see [70]), if it satisfies the condition (d)

–

𝑎 ∧ 𝑏 = 0 󳨐⇒ (𝑎𝑐) ∧ (𝑏𝑐) = (𝑐𝑎) ∧ (𝑐𝑏) = 0

Equivalent to (d) are the conditions: (d1 ) |𝑎𝑏| = |𝑎||𝑏| for all 𝑎, 𝑏 ∈ A, (d2 ) 𝑐(𝑎 ∧ 𝑏) = 𝑐𝑎 ∧ 𝑐𝑏 and (𝑎 ∧ 𝑏)𝑐 = 𝑎𝑐 ∧ 𝑏𝑐 for all 𝑎, 𝑏 ∈ A, 𝑐 ∈ A+ , (d3 ) 𝑐(𝑎 ∨ 𝑏) = 𝑐𝑎 ∨ 𝑐𝑏 and (𝑎 ∨ 𝑏)𝑐 = 𝑎𝑐 ∨ 𝑏𝑐 for all 𝑎, 𝑏 ∈ A, 𝑐 ∈ A+ ; see [21, Proposition 1.2]. An ℓ-algebra is called an almost 𝑓-algebra, if it satisfies the condition (ff)

–

for all 𝑐 ≥ 0.

𝑎 ∧ 𝑏 = 0 󳨐⇒ 𝑎𝑏 = 0.

Equivalent to (ff) is the condition 𝑎2 = |𝑎|2 for all 𝑎 ∈ A; see [21, Proposition 1.3]. An ℓ-algebra is called an 𝑓-algebra if it satisfies the condition (f)

𝑎 ∧ 𝑏 = 0 󳨐⇒ (𝑎𝑐) ∧ 𝑏 = (𝑐𝑎) ∧ 𝑏 = 0

for all 𝑐 ≥ 0.

Equivalent to (f) is the condition: (f1 ) – –

–

–

{𝑎𝑏}⊥⊥ ⊂ {𝑎}⊥⊥ ∩ {𝑏}⊥⊥

for

0 ≤ 𝑎, 𝑏 ∈ A;

see [103, Proposition 3.5]. An element 𝑒 ∈ A is called a multiplicative unit, if 𝑎 ⋅ 𝑒 = 𝑒 ⋅ 𝑎 = 𝑎 for all 𝑎 ∈ A. It is uniquely defined. An algebra with a multiplicative unit is called unitary. If A is an algebra with a multiplicative unit 𝑒, then an element 𝑎 ∈ A is called invertible, if a unique element 𝑏 ∈ A exists such that 𝑎𝑏 = 𝑏𝑎 = 𝑒. The element 𝑏 is denoted by 𝑎−1 . An element 𝑎 ∈ A is called nilpotent if there exists 𝑛 ∈ ℕ such that 𝑎𝑛 = 0. The set of all nilpotent elements of A is denoted by 𝑁(A). If A is an (Archimedean) 𝑓-algebra, then 𝑁(A) = {𝑎 ∈ A : 𝑎2 = 0}; see [103, Proposition 10.2 (i)]. An ℓ-algebra A is called semiprime, if the only nilpotent element in A is zero.

16 It is convenient to write 𝑎𝑏 instead of 𝑎 ⋅ 𝑏 for the product of 𝑎 and 𝑏.

3.5 Finite elements in ℓ-algebras and in product algebras |

51

In the following remarks we collect without proof the main properties of the introduced ℓ-algebras and comment on the relations between them. For the proofs we refer to [9, 21, 22, 95] and [143]. Remark 3.52. Let A be an arbitrary ℓ-algebra. (1) It follows immediately from the definitions that each 𝑓-algebra is a 𝑑-algebra. Each 𝑓-algebra is also an almost 𝑓-algebra. A 𝑑-algebra is not necessarily an almost 𝑓-algebra, nor vice versa. (2) If a 𝑑-algebra is semiprime or possesses a positive multiplicative unit, then it is an 𝑓-algebra. Each semiprime almost 𝑓-algebra is an 𝑓-algebra. (3) Each Archimedean commutative 𝑑-algebra is an almost 𝑓-algebra, and each Archimedean almost 𝑓-algebra is commutative. (4) Even in an 𝑓-algebra, the existence of a multiplicative unit is not guaranteed: The vector lattice c0 of all real zero sequences with the coordinatewise order and algebraic operations is a semiprime Archimedean 𝑓-algebra without multiplicative unit. (5) If, in an almost 𝑓-algebra, a multiplicative unit exists, then the latter is always positive. (6) Consider A = ℝ2 with the coordinatewise addition, scalar multiplication and partial order. If the multiplication is defined by (

𝑥1 (𝑦1 + 𝑦2 ) + 𝑥2 (𝑦1 + 12 𝑦2 ) 𝑥1 𝑦 ) ⋅ ( 1) = ( ), 1 𝑥2 𝑦2 𝑥2 𝑦2 2

then A is an ℓ-algebra, which is neither a 𝑑-algebra nor an almost 𝑓-algebra. It is clear that the nonpositive element 𝑒 = (−12) is the multiplicative unit in A; see [55]. (7) If a 𝑑-algebra or an Archimedean almost 𝑓-algebra A possesses a positive multiplicative unit, then A is an 𝑓-algebra. (8) An Archimedean ℓ-algebra with a multiplicative unit 𝑒 > 0 is an 𝑓-algebra if and only if 𝑒 is a weak order unit. (9) Every unitary Archimedean 𝑓-algebra is semiprime. (10) In an Archimedean commutative 𝑑-algebra for the vector lattice operations with 𝑝-th powers of 𝑎, 𝑏 ∈ A+ for 𝑝 ∈ ℕ≥1 , the following frequently used formulas hold (𝑎 ∧ 𝑏)𝑝 = 𝑎𝑝 ∧ 𝑏𝑝

and (𝑎 ∨ 𝑏)𝑝 = 𝑎𝑝 ∨ 𝑏𝑝 ;

(3.15)

see [26, Proposition 4], and [22, Proposition 1]). Throughout this section we consider only Archimedean ℓ-algebras, A, and vector lattices, 𝐸. A subalgebra of an algebra A is a vector subspace B ⊂ A, which is closed under the multiplication given in A. If 𝑢 ∈ B ⇒ 𝑢𝑎, 𝑎𝑢 ∈ B for any 𝑎 ∈ A, then B is called an (algebraic or) ring ideal. A ring ideal in A is an ℓ-ideal if it is simultaneously an order ideal in A. For the relations between order and ring ideals in 𝑓-algebras see [28, § 13].

52 | 3 Finite, totally finite and selfmajorizing elements The following scheme (Fig. 3.2) gives an overview of all implications between the various kinds of the introduced ℓ-algebras. The implications marked by a thick arrow are valid under the indicated condition.

(d) 𝑑-algebra

(f)

almost 𝑓-algebra

semiprime

co

(ff)

m -

m ut at iv e

ℓ-algebra

𝑓-algebra semiprime

Fig. 3.2. The relations between algebras.

3.5.2 Finite elements in unitary ℓ-algebras The first result shows that the multiplication with elements from the order ideal generated by the positive multiplicative unit preserves the finiteness of an element with the same majorant (and so also the total finiteness). Theorem 3.53. Let A be an ℓ-algebra with a positive multiplicative unit 𝑒 > 0, and let 𝑎 be an arbitrary element of 𝐼𝑒 = {𝑎 ∈ A : |𝑎| ≤ 𝜆𝑒 for some 𝜆 ∈ ℝ+ }. Then for 𝑖 = 1, 2 𝜑 ∈ Φ𝑖 (A) holds, with the majorant 𝑢 󳨐⇒ 𝜑𝑎, 𝑎𝜑 ∈ Φ𝑖 (A), with the majorant 𝑢. Proof. Without loss of generality, let 𝜑 ≥ 0 (otherwise use 𝜑 = 𝜑+ − 𝜑− ). It suffices to consider only 𝑎 ≥ 0, since by condition (ℓ2 ) there holds |𝑎𝜑| ≤ |𝑎|𝜑. For an element 𝑎 ∈ 𝐼𝑒 , there is a 𝜆 ∈ ℝ≥0 , such that 0 ≤ 𝑎 ≤ 𝜆𝑒. Due to the condition (ℓ1 ) we have for arbitrary 𝑥 ∈ A, and all 𝑛 ∈ ℕ, the inequality |𝑥| ∧ 𝑛 𝑎𝜑 ≤ |𝑥| ∧ 𝑛𝜆 𝑒𝜑 = |𝑥| ∧ 𝑛𝜆 𝜑. If now 𝜑 is a finite element with a majorant 𝑢, then |𝑥| ∧ 𝑛 𝑎𝜑 ≤ 𝑐𝑥 𝑢 for all 𝑛 ∈ ℕ. Therefore, the product 𝑎𝜑 is also a finite element with the same majorant as 𝜑. Analogously, the statement is proved for the product 𝜑𝑎.

3.5 Finite elements in ℓ-algebras and in product algebras

| 53

If 𝜑 is even totally finite, i. e., the majorant 𝑢 of 𝜑 itself is a finite element, then the products 𝑎𝜑 and 𝜑𝑎 also have finite majorants, which shows that they are totally finite as well. The same result can be proved without the positivity of the multiplicative unit, if A is supposed to be a 𝑑-algebra. However, in contrast to the previous theorem, the majorant for the product changes and depends on the factor 𝑎. Theorem 3.54. Let A be a 𝑑-algebra with a (not necessarily positive) multiplicative unit, and let 𝑎 ∈ A be an arbitrary element. Then for 𝑖 = 1, 2 there holds 𝜑 ∈ Φ𝑖 (A) 󳨐⇒ 𝑎𝜑, 𝜑𝑎 ∈ Φ𝑖 (A). In particular, Φ𝑖 (A) is a 𝑑-subalgebra and a ring ideal in A. If A, in addition, is an 𝑓-algebra, then Φ𝑖 (A) is even an 𝑓-subalgebra. Proof. Denote the multiplicative unit of A by 𝑒 and assume again 𝜑 ≥ 0. Let first 𝑖 = 1. Due to 𝜑 ∈ Φ1 (A), there is a majorant 𝑢 ∈ A for 𝜑, and for each 𝑥 ∈ A+ , a number 𝑐𝑥 ∈ ℝ+ , such that |𝑥| ∧ 𝑛 𝜑 ≤ 𝑐𝑥 𝑢 for all 𝑛 ∈ ℕ.

(3.16)

Since by condition (d1 ) one has |𝜑𝑎| = |𝜑||𝑎|, the elements 𝜑|𝑎| and 𝜑𝑎 are coincidentally finite, so 𝑎 ≥ 0 may be assumed. Let 𝑥 ∈ A and 𝑛 ∈ ℕ be arbitrary. Then 𝑎 ≥ 0 implies 𝑎 ∨ 𝑒 ≥ 0, and by means of condition (d3 ) from (3.16) (𝑎 ∨ 𝑒) |𝑥| ∧ 𝑛(𝑎 ∨ 𝑒)𝜑 ≤ 𝑐𝑥 (𝑎 ∨ 𝑒)𝑢 follows. Since 𝑎 ≤ 𝑎 ∨ 𝑒, and |𝑥| = 𝑒 |𝑥| ≤ (𝑎 ∨ 𝑒) |𝑥|, one has |𝑥| ∧ 𝑛 𝑎𝜑 ≤ |𝑥| ∧ 𝑛(𝑎 ∨ 𝑒)𝜑 ≤ (𝑎 ∨ 𝑒) |𝑥| ∧ 𝑛(𝑎 ∨ 𝑒)𝜑 ≤ 𝑐𝑥 (𝑎 ∨ 𝑒)𝑢 for all 𝑛 ∈ ℕ, i. e., the element 𝑎𝜑 is finite in A with the majorant (𝑎 ∨ 𝑒)𝑢. Analogously it will be shown that the product 𝜑𝑎 is finite in A. The set Φ1 (A) is an order ideal in A, in particular a vector sublattice. According to the first part of the proof, the product of two finite elements is finite and thus the set Φ1 (A) is closed under the multiplication. The properties (d) or (f) are shared by the set Φ1 (A), if A is a 𝑑- or an 𝑓-algebra respectively. Therefore, Φ1 (A) is a 𝑑- or an 𝑓-subalgebra of A respectively. It is clear from the proof that Φ1 (A) is a ring ideal. For 𝑖 = 2, observe that (𝑎 ∨ 𝑒)𝑢 is a majorant of the element 𝑎𝜑 as above, where 𝑢 as a majorant of the totally finite element 𝜑 can be assumed to be a finite element. By what has been proved in the case 𝑖 = 1, the element (𝑎 ∨ 𝑒)𝑢 is finite as well, which yields the total finiteness of 𝑎𝜑 in A. The total finiteness of the product 𝜑𝑎 is proved analogously. The remaining statements for Φ2 (A) are obtained analogously to the case 𝑖 = 1.

54 | 3 Finite, totally finite and selfmajorizing elements Remark 3.55. If 𝑢 is a majorant of 𝜑, then a majorant of 𝑎𝜑, 𝜑𝑎 is (𝑎 ∨ 𝑒)𝑢, 𝑢(𝑎 ∨ 𝑒) respectively. In particular, the idea of the proof cannot be used to obtain an analogous result for selfmajorizing elements. If the multiplicative unit itself is a finite element we get the following theorem. Theorem 3.56. Let A be a 𝑑-algebra with a multiplicative unit 𝑒. Let 𝑒 ∈ Φ1 (A). Then Φ1 (A) = Φ2 (A) = A. If A is an 𝑓-algebra and 𝑒 ∈ Φ1 (A), then 𝑒 is even an order unit in A and Φ1 (A) = Φ2 (A) = Φ3 (A) = A. Proof. First consider the case of a 𝑑-algebra. Since 𝑒 is finite, by the previous theorem the products 𝑎𝑒 and 𝑒𝑎 for all 𝑎 ∈ A are finite elements as well, i. e., A ⊆ Φ1 (A). So the equalities Φ1 (A) = Φ2 (A) = A hold. Consider the case of an 𝑓-algebra¹⁷ A. We mention first that 𝑒 is positive, as any multiplicative unit in an almost 𝑓-algebra (Remark 3.52 (5)). Then, by Theorem 1.9 of [21], the multiplicative unit 𝑒 is a weak order unit. If A has a weak order unit then, according to Corollary 3.10, the equalities Φ1 (A) = Φ2 (A) = A hold if and only if an order unit in A exists. Since the equalities hold according to what was proved in the first part (here we use the fact that an 𝑓-algebra is also a 𝑑-algebra), the 𝑓-algebra A has an order unit. From Proposition 3.44 it now follows that A also coincides with the order ideal Φ3 (A) of the selfmajorizing elements of A. Now consider the weak order unit 𝑒 which, due to A = Φ3 (A), is a selfmajorizing element, and show that 𝑒 is a (strong) order unit. According to Remark 3.43, the ideal generated in A by 𝑒 is a projection band and coincides with {𝑒}⊥⊥. Since 𝑒 is a weak order unit, one has A = {𝑒}⊥⊥. Consequently, A = {𝑒}⊥⊥ = {𝑎 ∈ A : ∃𝜆 > 0 with |𝑎| ≤ 𝜆𝑒}, i. e., 𝑒 is an order unit in A. A Riesz norm ‖⋅‖ on an ℓ-algebra A is called a submultiplicative Riesz norm, if ‖𝑎𝑏‖ ≤ ‖𝑎‖ ‖𝑏‖ ,

whenever 𝑎, 𝑏 ∈ A+ .

Theorem 3.57. Let A be an almost 𝑓-algebra with a multiplicative unit 𝑒. Let a submultiplicative Riesz norm exist on A. Then (1) the multiplicative unit 𝑒 is an order unit; and (2) Φ1 (A) = Φ2 (A) = Φ3 (A) = A.

17 In view of Remark 3.52 (1) then A, in particular, is both an almost 𝑓-algebra and a 𝑑-algebra. However, it is not known whether the statement is true for an almost 𝑓-algebra A where, due to Remarks 3.52 (5) and (7), only the non-Archimedean case is of interest.

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Proof. (1) We show¹⁸ that for each 𝑎 ∈ A, 0 ≠ 𝑎 there is a 𝜆 ∈ ℝ+ such that −𝜆𝑒 ≤ 𝑎 ≤ 𝜆𝑒. Let first 𝑎 ∈ A+ . Further on, the obvious decomposition 𝑎 − 𝜆𝑒 = (𝑎 − 𝜆𝑒)+ − (𝑎 − 𝜆𝑒)− is used, which holds for any 𝜆 ∈ ℝ+ . Now consider the element (𝑎 − 𝜆𝑒)(𝑎 − 𝜆𝑒)+. The case (𝑎 − 𝜆𝑒)+ = 0 for some 𝜆 > 0 leads to 𝑎 − 𝜆𝑒 = −(𝑎 − 𝜆𝑒)− ≤ 0, so 0 ≤ 𝑎 ≤ 𝜆𝑒, and we are done. Therefore we deal with the case (𝑎 − 𝜆𝑒)+ > 0 and consider the element (𝑎 − 𝜆𝑒)(𝑎 − 𝜆𝑒)+ . Due to the condition (ff), the product of the two positive disjoint elements (𝑎 − 𝜆𝑒)− and (𝑎 − 𝜆𝑒)+ vanishes, and by taking the condition (ℓ) into account we obtain the inequality (𝑎−𝜆𝑒)(𝑎−𝜆𝑒)+ = 2 (𝑎 − 𝜆𝑒)+ (𝑎 − 𝜆𝑒)+ − (𝑎 − 𝜆𝑒)− (𝑎 − 𝜆𝑒)+ = ((𝑎 − 𝜆𝑒)+) ≥ 0. We conclude 𝑎(𝑎 − 𝜆𝑒)+ − 𝜆𝑒 (𝑎 − 𝜆𝑒)+ ≥ 0, and so 𝑎(𝑎 − 𝜆𝑒)+ ≥ 𝜆(𝑎 − 𝜆𝑒)+ > 0 for 𝜆 > 0. Due to the norm being submultiplicative and Riesz, we obtain 󵄩 󵄩 󵄩 +󵄩 +󵄩 +󵄩 ‖𝑎‖ 󵄩󵄩󵄩(𝑎 − 𝜆𝑒) 󵄩󵄩󵄩 ≥ 󵄩󵄩󵄩𝑎(𝑎 − 𝜆𝑒) 󵄩󵄩󵄩 ≥ 𝜆 󵄩󵄩󵄩(𝑎 − 𝜆𝑒) 󵄩󵄩󵄩 > 0, and therefore 𝜆 ≤ ‖𝑎‖. Altogether, as we have seen, the assumption (the unfavourable case) (𝑎 − 𝜆𝑒)+ > 0 leads to 𝜆 ≤ ‖𝑎‖. Therefore all 𝜆 > ‖𝑎‖ yield to (the favourable case) (𝑎 − 𝜆𝑒)+ = 0, and so 𝑎 − 𝜆𝑒 = −(𝑎 − 𝜆𝑒)− ≤ 0, and again 0 ≤ 𝑎 ≤ 𝜆𝑒 as above. Now let 𝑎 ∈ A be an arbitrary element. In view of ±𝑎 ≤ |𝑎|, we obtain the claimed result. (2) The fact that all elements in A are finite, totally finite, and even selfmajorizing follows by taking into account that 𝑒 is an order unit in A, and by applying Proposition 3.44. Notice that the existence of a Riesz norm on an algebra A implies that A is Archimedean. Therefore, with respect to Remarks 3.52 (5) and (7), there are actually 𝑓-algebras considered in the theorem. It is well known that the collection Orth(𝐸) of all orthomorphisms on an Archimedean vector lattice 𝐸 is an 𝑓-algebra with the identity of a weak order unit. Moreover, any 𝑓-algebra A with a multiplicative unit 𝑒 is algebraic and lattice-isomorphic to Orth(A), where the image of 𝑒 is the identity in Orth(A) ([95, Theorems 3.1.10 and 3.1.13]). Corollary 3.58. Let A be a unitary 𝑓-algebra. Let a submultiplicative Riesz norm exist on A. Then (1) the identity operator 𝐼 is an order unit in Orth(A); and (2) Orth(A) = Φ𝑖 (Orth(A)), 𝑖 = 1, 2, 3. A similar result holds if there is some norm on the algebra A, which turns it into a Banach lattice.

18 The main idea of the proof is based on W. A. J. Luxemburg [83], cf. [9, Theorem 15.5].

56 | 3 Finite, totally finite and selfmajorizing elements Theorem 3.59. Let A be an 𝑓-algebra with a multiplicative unit 𝑒. Let a norm exist on A such that A becomes a Banach lattice. Then (1) the multiplicative unit 𝑒 is an order unit; and (2) Φ1 (A) = Φ2 (A) = Φ3 (A) = A. Proof. Since the 𝑓-algebras A and Orth(A) are algebraic and lattice isomorphic, such that the image of 𝑒 under the isomorphism is 𝐼 ∈ Orth(A), then according to Wickstead’s Theorem ([9, Theorem.15.5]), the identity operator 𝐼 is an order unit in Orth(A), and so 𝑒 is an order unit in A. By virtue of Proposition 3.44, all elements in Orth(A), and consequently in A, are finite, totally finite and even selfmajorizing. For the 𝑓-algebra of all orthomorphisms on a vector lattice we get the following properties from Theorems 3.54 and 3.59, which we formulate as follows. Corollary 3.60. (1) Let 𝐸 be a vector lattice. If 𝑆 ∈ Φ𝑖 (Orth(𝐸)) for 𝑖 = 1, 2, and 𝑇 ∈ Orth(𝐸), then also 𝑆 ∘ 𝑇 ∈ Φ𝑖 (Orth(𝐸)). In particular, Φ𝑖 (Orth(𝐸)) is an 𝑓-subalgebra and a ring ideal. (2) Let 𝐸 be a Banach lattice. Then Orth(𝐸) is an 𝑓-algebra, and under the order unit norm ‖𝑇‖𝐼 = inf{𝜆 > 0 : |𝑇| ≤ 𝜆𝐼} also an 𝐴𝑀-space with order unit. Then by the previous theorem we have Orth(𝐸) = Φ𝑖 (Orth(𝐸)),

𝑖 = 1, 2, 3.

The last results also throw some light on the relationship between finiteness and invertibility of elements in 𝑓-algebras. Example 3.61. Consider the vector lattice 𝐶𝑏 (ℝ) of all bounded real-valued continuous functions on ℝ equipped with the pointwise algebraic operations and partial order. Then 𝐶𝑏 (ℝ) turns out to be an Archimedean 𝑓-algebra with multiplicative units, Φ1 (A) ≠ {0} and no finite element is invertible. 󵄩 󵄩 𝐶𝑏 (ℝ) is a Banach lattice if the norm is defined by 󵄩󵄩󵄩𝑓󵄩󵄩󵄩∞ = sup𝑥∈ℝ |𝑓(𝑥)| for 𝑓 ∈ 𝐶𝑏 (ℝ). Since (many) order units exist in 𝐶𝑏 (ℝ) (e. g., the function 1), all elements are finite¹⁹. Observe that any function 𝑓 ∈ 𝐶𝑏(ℝ) with inf 𝑥∈ℝ |𝑓(𝑥)| > 0 is invertible. Of course, there are noninvertible elements as well, e. g., the functions with compact support. Let A be a 𝑑-algebra with a multiplicative unit 𝑒. If at least one nonzero finite element exists which is invertible in A, then immediately all elements of A are finite, i. e., A = Φ1 (A). Indeed, if an element 𝜑 is both finite and invertible, then Theorem 3.54 guarantees that the element 𝑒 = 𝜑−1 𝜑 is finite in A. Then by Theorem 3.56 all elements of A are finite. The 𝑓-algebra 𝐶(ℝ) of all continuous functions on ℝ contains a multiplicative unit (the function 1); however, in contrast to 𝐶𝑏 (ℝ) there is no order unit. There is also no

19 The Banach algebras 𝐶𝑏 (ℝ) and 𝐶(𝛽ℝ) are lattice isomorphic, where 𝛽ℝ denotes the Stone–Čech compactification of ℝ, see [8]. Therefore all elements in 𝐶𝑏 (ℝ), as in 𝐶(𝛽ℝ), are finite.

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norm on 𝐶(ℝ), which makes it a Banach lattice. Otherwise, by Theorem 3.59, there would be an order unit. It is clear that the element 1 is not finite²⁰ in 𝐶(ℝ). By what has been mentioned above, no finite element can be invertible. Consequently, there exist 𝑓-algebras A with multiplicative units, such that Φ1 (A) ≠ {0} and no finite element is invertible.

3.5.3 Finite elements in nonunitary 𝑓-algebras In this section we consider 𝑓-algebras which do not possess any multiplicative unit. Definition 3.62. Let A be an 𝑓-algebra. (i) A is said to be square-root closed if for any 𝑎 ∈ A+ an element 𝑏 ∈ A exists such that 𝑏2 = 𝑎, i. e., for any such element 𝑎 a square root 𝑏 exists. (ii) A is said to have the factorization property if for every 𝑎 ∈ A two elements 𝑏, 𝑐 ∈ A exist such that 𝑎 = 𝑏𝑐. (iii) A is said to have the weak factorization property if for every 𝑎 ∈ A two elements 𝑏, 𝑐 ∈ A exist such that 𝑎 ≤ 𝑏𝑐. In [23, Theorem 4.6], the first two properties were proved to be equivalent in uniformly complete 𝑓-algebras. The fact that the property (iii) is weaker than (ii) is demonstrated in the next example. Example 3.63. An example of a uniformly complete 𝑓-algebra which satisfies the weak factorization property but fails to have the factorization property. For the vector lattice A = {𝑓 ∈ 𝐶[−1, 1] : 𝑓(0) = 0} let the multiplicationfor all 𝑓, 𝑔 ∈ A be defined by {𝑓(𝑡)𝑔(𝑡), 𝑡 ∈ [0, 1] (𝑓 ⋅ 𝑔)(𝑡) = { 𝑓(−𝑡)𝑔(−𝑡), 𝑡 ∈ [−1, 0). { Products in A are precisely axisymmetric functions which vanish at 0. Observe that A is an 𝑓-algebra which is not semiprime. We will show that A is uniformly complete and has the weak factorization property. However, the factorization property does not hold in A. To see that A is uniformly complete, notice that A is the kernel 𝛿0−1 (0) of the continuous functional 𝛿0 defined on the Banach lattice 𝐶[−1, 1] by 𝛿0 (𝑓) = 𝑓(0). The 𝑓-algebra A obviously does not possess the factorization property, since an arbitrary 𝑔 ∈ A, which is not axisymmetric, cannot be written as a product of two elements of A. Since in uniformly complete 𝑓-algebras the factorization property is equivalent to the square-root closedness, the latter does not hold in A either. However, A has the weak

20 Since Φ1 (𝐶(ℝ)) = K(ℝ) is the vector lattice of all functions with compact support.

58 | 3 Finite, totally finite and selfmajorizing elements factorization property. Indeed, let 𝑔 ∈ A be an arbitrary element. Define ̂ := max {|𝑔(𝑡)|, |𝑔(−𝑡)|} 𝑔(𝑡) 𝑡∈[−1,1]

and

̂ ̃ := √𝑔(𝑡). 𝑔(𝑡)

Then 𝑔,̂ 𝑔 ̃ ∈ A and the relation 𝑔 ≤ 𝑔 ̂ = 𝑔 ̃ 2 holds. Example 3.64. An example of a semiprime 𝑓-algebra which does not satisfy the weak factorization property. Consider finite partitions t of the set [0, ∞) into subintervals 𝐼𝑘 of [0, ∞) for any 𝑘, i. e., t = {𝐼0 , . . . , 𝐼𝑛 } such that 𝑛

⋃ 𝐼𝑘 = [0, ∞), where 𝐼𝑘 ∩ 𝐼𝑗 = 0 for 𝑘 ≠ 𝑗. 𝑘=0

Let ℙ0 be the set of all polynomials 𝑝 vanishing at the point 𝑡 = 0. Consider now the collection P = P([0, ∞)) of all continuous (piecewise polynomials) functions on [0,∞), 󵄨 for each of which a partition t exists such that 𝑓󵄨󵄨󵄨𝐼 = 𝑝𝑘 with 𝑝𝑘 ∈ ℙ0 for any 𝐼𝑘 ∈ 𝑘 t. The algebraic operations, the multiplication, and the partial order are introduced in P pointwise. Then P is an Archimedean ℓ-algebra. Moreover, it is easy to see that the disjointness²¹ of two functions 𝑓, 𝑔 ∈ P is also preserved after the multiplication of one of them by a positive function ℎ ∈ P. Therefore P is an 𝑓-algebra. Since only the zeroelement of P can satisfy the equation 𝑓2 = 0, the 𝑓-algebra is semiprime. Observe that the restriction on [0, ∞) of a polynomial 𝑝 of arbitrary degree with 𝑝(0) = 0 belongs to 󵄨 P, but the function 1󵄨󵄨󵄨[0,∞) does not. It follows that P does not contain either an order unit or a multiplicative unit. The 𝑓-algebra P does not possess the weak factorization property, since the function 𝑓(𝑡) = 𝑡 cannot be estimated by a product of two functions. Indeed, 𝑓 ≤ 𝑝𝑞 implies that both polynomials 𝑝, 𝑞 take on positive values for all 𝑡 > 0 and deg(𝑝𝑞) ≥ 2. Since 𝑝𝑞(0) = 0, the graphs of 𝑓 and 𝑝𝑞 intersect at some point. Let 𝑡0 be the smallest number with 0 < 𝑡0 , and 𝑓(𝑡0 ) = 𝑝𝑞(𝑡0). There is an interval 𝐼𝑘 of a partition for 𝑝𝑞 ∈ P such that 𝑡0 ∈ 𝐼𝑘 , and 𝑓(𝑡) > 𝑝𝑞(𝑡) for 𝑡 ∈ (0, 𝑡0 ). For the product of the 𝑝 elements 𝑔1 , . . . , 𝑔𝑝 in an ℓ-algebra A we will use the notation 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 := 𝑔1 ⋅ . . . ⋅ 𝑔𝑝 . The 𝑝-fold product²² is used to define the 𝑝-th root of an element in an ℓ-algebra as follows: for 𝑔 ∈ A, an element 𝑔 ̃ ∈ A is called a 𝑝-th root of 𝑔 if 1 𝑔𝑝̃ = 𝑔. If a 𝑝-th root of 𝑔 exists and is uniquely defined, then we write 𝑔̃ = 𝑔 𝑝 and call 𝑔 ̃ the 𝑝-th root of 𝑔. For details we refer to [22], [27] and [143], where in particular the following results can be found.

21 The supports of two disjoint continuous functions on [0, ∞) intersect at most at one point. 22 As usual, for the element 𝑔 ∈ A, its 𝑝-fold product 𝑔 ⋅ ⋅ ⋅ 𝑔 is denoted by 𝑔𝑝 .

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Remark 3.65. (1) Existence and uniqueness of the root. Let A be an Archimedean uniformly complete almost 𝑓-algebra, and 𝑝 ∈ ℕ≥2 . Then there exists a positive 𝑝-th root of any 𝑝-fold product of positive elements of A, i. e., 1 𝑔1 , . . . , 𝑔𝑝 ∈ A+ 󳨐⇒ (𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 ) 𝑝 exists in A+ , (3.17) see [26, Theorem 3]. The root is uniquely defined if the algebra A is semiprime. In this case, according to Remark 3.52 (2), A is an 𝑓-algebra. (2) Monotonicity of the root. In every ℓ-algebra A for 𝑝 ∈ ℕ≥2 and 𝑎, 𝑏 ∈ A+ , due to the property (ℓ1 ), one has²³ 𝑎 ≤ 𝑏 󳨐⇒ 𝑎𝑝 ≤ 𝑏𝑝 . If A is a semiprime 𝑓-algebra, then the root is monotone, i. e., 𝑎 ≤ 𝑏 ⇐⇒ 𝑎𝑝 ≤ 𝑏𝑝, see [143, Theorem 142.3], and [22, Proposition 2 (iii)]. Theorem 3.66. Let A be a uniformly complete 𝑓-algebra with the weak factorization property, 𝑝 ∈ ℕ≥2 and 𝑖 = 1, 2. If 𝜑1 , . . . , 𝜑𝑝 ∈ Φ𝑖 (A) with majorants 𝑢1 , . . . , 𝑢𝑝 respectively, then 𝜑1 ⋅ ⋅ ⋅ 𝜑𝑝 ∈ Φ𝑖 (A) with a majorant (𝑢1 ∨ . . . ∨ 𝑢𝑝 )𝑝. In particular, Φ𝑖 (A) is an 𝑓-subalgebra of A. Proof. Let 𝑖 = 1. First, we prove the claim for the 𝑝-fold power 𝜑𝑝 of a finite element 𝜑 ∈ A. Let 𝜑 be a finite element in A with a majorant 𝑢 ∈ A+ . Without loss of generality, 𝜑 can be assumed to be positive, otherwise use |𝜑𝑝 | = |𝜑|𝑝 , which holds due to property (d1 ). For an arbitrary 𝑎 ∈ A, the weak factorization property of A yields the existence of 𝑝 elements 𝑎1 , . . . , 𝑎𝑝 ∈ A, with |𝑎| ≤ 𝑎1 ⋅ ⋅ ⋅ 𝑎𝑝 . According to property (ℓ2 ) and Remark 3.65 (1), there follows the positivity of the elements 𝑎1 , . . . , 𝑎𝑝 , and the existence 1

of a root (𝑎1 ⋅ ⋅ ⋅ 𝑎𝑝 ) 𝑝 in A. Using the Formula (3.15) and the finiteness of 𝜑, we obtain that there is a constant 𝑐√𝑝 𝑎1 ⋅⋅⋅𝑎𝑝 ≥ 0 such that for all 𝑛 ∈ ℕ 1

𝑝

𝑝

|𝑎| ∧ 𝑛𝜑𝑝 ≤ (𝑎1 ⋅ ⋅ ⋅ 𝑎𝑝 ) ∧ 𝑛𝜑𝑝 = ((𝑎1 ⋅ ⋅ ⋅ 𝑎𝑝 ) 𝑝 ∧ 𝑛𝜑) ≤ 𝑐√𝑝 𝑎 ⋅⋅⋅𝑎 𝑢𝑝 , 1

𝑝

where the last inequality follows from Remark 3.65 (2). Therefore the 𝑝-th power 𝜑𝑝 of a finite element 𝜑 ∈ A is also finite.

23 The twofold application of condition (ℓ1 ) to 0 ≤ 𝑎 ≤ 𝑏 yields 𝑎2 ≤ 𝑏2 . Indeed, by multiplying the inequality 0 ≤ 𝑎 ≤ 𝑏 with 𝑎, and respectively 𝑏, one obtains 𝑎2 ≤ 𝑎𝑏, respectively 𝑎𝑏 ≤ 𝑏2 .

60 | 3 Finite, totally finite and selfmajorizing elements Now let 𝜑1 , . . . , 𝜑𝑝 be arbitrary finite elements in A with majorants 𝑢1 , . . . , 𝑢𝑝 , respectively. The modulus of the product 𝜑1 ⋅ ⋅ ⋅ 𝜑𝑝 can be estimated by |𝜑1 ⋅ ⋅ ⋅ 𝜑𝑝 | ≤ |𝜑1 | ⋅ ⋅ ⋅ |𝜑𝑝 | ≤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (|𝜑1 | ∨ ⋅ ⋅ ⋅ ∨ |𝜑𝑝 |) ⋅ ⋅ ⋅ (|𝜑1 | ∨ ⋅ ⋅ ⋅ ∨ |𝜑𝑝 |) 𝑝 times

= (|𝜑1 | ∨ ⋅ ⋅ ⋅ ∨ |𝜑𝑝 |)𝑝 . Since a majorant of the supremum |𝜑1 | ∨ ⋅ ⋅ ⋅ ∨ |𝜑𝑝 | is given by 𝑢1 ∨ . . . ∨ 𝑢𝑝 , the 𝑝fold product (|𝜑1 | ∨ ⋅ ⋅ ⋅ ∨ |𝜑𝑝 |)𝑝 is finite as well, by the first part with the majorant (𝑢1 ∨ . . . ∨ 𝑢𝑝 )𝑝 . Let 𝑖 = 2. In this case, the majorants 𝑢1 , . . . , 𝑢𝑝 are assumed to belong to Φ1 (A), and the element (𝑢1 ∨ . . . ∨ 𝑢𝑝 )𝑝 is finite due to what was proved in case 𝑖 = 1. The last theorem was proved under stronger conditions than we will use in Theorem 3.69, where we drop the uniform completeness and the weak factorization property of the 𝑓-algebra. However, in the proof of Theorem 3.69, the majorants are not given explicitly and so, in contrast to Theorem 3.66, the fate of totally finite elements remains unknown there. For the next theorem notice that Example 3.63 shows that the weak factorization property does not imply semiprimitivity, even under the additional condition of uniform completeness. According to Example 3.64, the converse implication is also not true. However, it is not known if the combination of uniform completeness and semiprimitivity imply the weak factorization property. Theorem 3.67. Let A be a semiprime uniformly complete 𝑓-algebra with the weak fac1 1 torization property and 𝑝 ∈ ℕ≥2 . If, for 𝜑 ∈ Φ1 (A) the root 𝜑 𝑝 exists in A, then 𝜑 𝑝 ∈ Φ1 (A). 1

Proof. First consider 0 < 𝜑 ∈ Φ1 (A) with a majorant 𝑢 ∈ A, for which the root 𝜑 𝑝 exists in A+ . Let 𝑎 ∈ A+ be an arbitrary element. According to Formula (3.15) and using the finiteness of 𝜑 we get 1 (𝑎 ∧ 𝑛𝜑 𝑝 )𝑝 = 𝑎𝑝 ∧ 𝑛𝑝 𝜑 ≤ 𝑐𝑎𝑝 𝑢 for some constant 𝑐𝑎𝑝 ≥ 0, and all 𝑛 ∈ ℕ. Due to the weak factorization property, there are 𝑝 elements 𝑢1 , . . . , 𝑢𝑝 ∈ A, such that 𝑢 ≤ 𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 . Therefore the above inequality can be continued as follows: 1

(𝑎 ∧ 𝑛𝜑 𝑝 )𝑝 ≤ 𝑐𝑎𝑝 𝑢 ≤ 𝑐𝑎𝑝 𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 . Due to condition (ℓ2 ), the relations 0 ≤ 𝑢 ≤ 𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 = |𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 | ≤ |𝑢1 | ⋅ ⋅ ⋅ |𝑢𝑝 | hold. 󵄨 󵄨 Without loss of generality we may replace 𝑢𝑖 by 󵄨󵄨󵄨𝑢𝑖 󵄨󵄨󵄨, and assume that 𝑢𝑖 ≥ 0 for 𝑖 = 1

1, . . . , 𝑝. According to Remark 3.65 (1), the root (𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 ) 𝑝 exists in A and we obtain 1

1

(𝑎 ∧ 𝑛𝜑 𝑝 )𝑝 ≤ 𝑐𝑎𝑝 𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 = 𝑐𝑎𝑝 ((𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 ) 𝑝 )𝑝 .

3.5 Finite elements in ℓ-algebras and in product algebras

61

|

The monotonicity of the root allows us to extract the 𝑝-th root on both sides under the preservation of the inequality, which yields 1

1

𝑝 𝑝 𝑎 ∧ 𝑛𝜑 𝑝 ≤ √𝑐 𝑎𝑝 (𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 ) , 1

1

which shows that the element 𝜑 𝑝 is finite in A with the majorant (𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 ) 𝑝 . 1

Now let 𝜑 ∈ Φ1 (A) be arbitrary. If 𝜑 possesses a root 𝜑 𝑝 , then by condition (d1 ) 1

1

1

1

(𝑝 times)

|𝜑| = |𝜑 𝑝 ⋅ ⋅ ⋅ 𝜑 𝑝 | = |𝜑 𝑝 | ⋅ ⋅ ⋅ |𝜑 𝑝 |, 1

1

1

implies that |𝜑 𝑝 | is a 𝑝-th root of |𝜑|, i. e., |𝜑 𝑝 | = |𝜑| 𝑝 . Together with 𝜑, the element |𝜑| 1

is also finite in A and so, according to the first part of the proof, the element |𝜑 𝑝 | is 1 finite, and therefore the finiteness of 𝜑 𝑝 is obtained. In analogy to the above theorem we obtain the next result. Corollary 3.68. Let A be a semiprime 𝑓-algebra, and 𝑝 ∈ ℕ≥2 . If for 𝜑 ∈ Φ3 (A) the root 1

1

𝜑 𝑝 exists in A, then 𝜑 𝑝 ∈ Φ3 (A). 1

Proof. First consider 0 < 𝜑 ∈ Φ3 (A), for which the root 𝜑 𝑝 exists in A+ . According to Formula (3.15), and since 𝜑 is selfmajorizing, we get 1

(𝑎 ∧ 𝑛𝜑 𝑝 )𝑝 = 𝑎𝑝 ∧ 𝑛𝑝 𝜑 ≤ 𝑐𝑎𝑝 𝜑 for some constant 𝑐𝑎𝑝 ≥ 0 and all 𝑛 ∈ ℕ. The monotonicity of the root allows us to extract the 𝑝-th root on both sides, which yields 1

1

𝑝 𝑝 𝑎 ∧ 𝑛𝜑 𝑝 ≤ √𝑐 𝑎𝑝 𝜑 , 1

and shows that the element 𝜑 𝑝 is selfmajorizing. 1 Now let 𝜑 ∈ Φ3 (A) be arbitrary such that 𝜑 𝑝 exists. The application of the iden1

1

1

tity |𝜑 𝑝 | = |𝜑| 𝑝 analogously to the proof of the previous theorem ensures that 𝜑 𝑝 is selfmajorizing. For the next result, which is similar to Theorem 3.54, we use the characterization of an 𝑓-algebra by condition (f1 ). Theorem 3.69. Let A be an 𝑓-algebra, 𝜑 ∈ Φ1 (A), and 𝑎 ∈ A. Then 𝑎𝜑 ∈ Φ1 (A). In particular, Φ1 (A) is an 𝑓-subalgebra and a ring ideal. Proof. Let first 𝑎 ∈ A+ and 𝜑 ∈ Φ1 (A), 𝜑 ≥ 0. Using the condition (f1 ) we obtain ⊥⊥ ⊥⊥ ⊥⊥ ⊆ {𝑎}⊥⊥ ∩{𝜑} . By Theorem 3.9 for the finite element 𝜑 we have {𝜑} ⊆ Φ1 (A), {𝑎𝜑} and so ⊥⊥ ⊥⊥ {𝑎𝜑} ⊆ {𝑎}⊥⊥ ∩ {𝜑} ⊆ Φ1 (A). In particular, the product 𝑎𝜑 is finite in A.

62 | 3 Finite, totally finite and selfmajorizing elements Now let 𝑎 ∈ A be arbitrary and 𝜑 positive. The first part of the proof yields 𝑎+ 𝜑, 𝑎− 𝜑 ∈ Φ1 (A), and so we obtain the finiteness of 𝑎𝜑 = 𝑎+ 𝜑 − 𝑎− 𝜑 in A. Finally, assume 𝜑 to be arbitrary. Since Φ1 (A) is an ideal, we obtain the finiteness of 𝜑+ and 𝜑− , and therefore also the finiteness of 𝑎𝜑 = 𝑎𝜑+ − 𝑎𝜑− in A. Note that the product 𝑎1 ⋅ ⋅ ⋅ 𝑎𝑝 belongs to Φ1 (A) if at least one of the elements 𝑎1 , . . . , 𝑎𝑝 ∈ A belongs to Φ1 (A). The next theorem generalizes Theorem 3.59 since, as was already mentioned in Remark 3.52 (9), a unitary 𝑓-algebra A is automatically semiprime. For its proof we need the following result, which we obtain by resuming and restricting Theorem 12.3.8 from [24]. First we introduce the following notation. Let A be an ℓ-algebra and 𝑐 ∈ A. Denote by 𝑐 𝜋 and 𝜋𝑐 the left and right multiplications by 𝑐 respectively, i. e., 𝑐 𝜋, 𝜋𝑐 : A → A, defined by 𝑐 𝜋(𝑎) = 𝑐 𝑎 , and 𝜋𝑐 (𝑎) = 𝑎 𝑐 for all 𝑎 ∈ A. It is clear that each multiplication operator 𝑐 𝜋, 𝜋𝑐 is order bounded. If A additionally satisfies the condition (f), then for each 𝑐 ≥ 0 the operators 𝑐 𝜋 and 𝜋𝑐 are bandpreserving (and hence orthomorphisms), since one then has 𝜋𝑐 (𝑎) ∧ 𝑏 = 𝑐 𝜋(𝑎) ∧ 𝑏 = 0, whenever 𝑎 ∧ 𝑏 = 0 (see [9, Theorem 8.2]). If the 𝑓-algebra A is Archimedean, then 𝑐 𝜋 = 𝜋𝑐 for each 𝑐. Notice that the map ℎ : 𝑎 󳨃→ 𝜋𝑎 from a 𝑑-algebra A into Orth(A) is a lattice homomorphism. Indeed, the condition (d2 ) implies 𝜋𝑎∧𝑏 (𝑐) = (𝑎 ∧ 𝑏)𝑐 = 𝑎𝑐 ∧ 𝑏𝑐 = 𝜋𝑎 (𝑐) ∧ 𝜋𝑏 (𝑐) = (𝜋𝑎 ∧ 𝜋𝑏 )(𝑐), and thus ℎ(𝑎 ∧ 𝑏) = ℎ(𝑎) ∧ ℎ(𝑏). The other properties of ℎ follow analogously. Proposition 3.70. For an Archimedean 𝑓-algebra A the following conditions are equivalent: (1) the algebra A is semiprime; (2) the map ℎ is an injective homomorphism from A into Orth(A). In particular, A is embeddable as an 𝑓-subalgebra into the Archimedean unitary 𝑓-algebra Orth(A). Proof. (1) ⇒ (2). Since A is semiprime, one has 𝜋𝑎 ≠ 0 for all 𝑎 ∈ A, 0 ≠ 𝑎. Therefore ker(ℎ) = {0}, i. e., ℎ is injective. (2) ⇒ (1). Since A is embeddable into Orth(A) by means of ℎ, we can identify A with a sublattice of Orth(A). If the element 𝑎 is nilpotent in A, then 𝑎 is also nilpotent in Orth(A). But the unitary 𝑓-algebra Orth(A) is semiprime, i. e., 𝑎 is the zero element in Orth(A) and also in A. Remark 3.71. Let A be a semiprime 𝑓-algebra. Then 𝜋𝜑 ∈ Φ3 (Orth(A)) 󳨐⇒ 𝜑 ∈ Φ3 (A). Indeed, by the previous proposition we can identify A with a sublattice of Orth(A), and so for each 𝑥 ∈ A we obtain 󵄨 󵄨 |𝑥| ∧ 𝑛|𝜑| = |𝜋𝑥 | ∧ 𝑛|𝜋𝜑 | ≤ 𝑐|𝜋𝑥 | |𝜋𝜑 | = 𝑐|𝜋𝑥 | 󵄨󵄨󵄨𝜑󵄨󵄨󵄨

3.5 Finite elements in ℓ-algebras and in product algebras |

63

for any 𝑛 ∈ ℕ and some constant 𝑐|𝜋𝑥 | ∈ ℝ+ . Notice that the same statement for finite and totally finite elements, in general, is not true, since in these cases the majorants might not belong to A. The inverse implication, in general, is not true because for 𝜑 ∈ Φ3 (A) the element 𝜋𝜑 ∈ Orth(A) may not be a majorant for itself. Indeed, if A does not possess any multiplicative unit, then for 𝑥 ∈ Orth(A) \ A a corresponding constant 𝑐𝑥 might not exist. Theorem 3.72. Let A be a semiprime 𝑓-algebra and let a norm exist on A, under which A is a Banach lattice. Then Φ1 (A) = Φ2 (A) = Φ3 (A) = A. Proof. Since A is semiprime, according to Proposition 3.70 the 𝑓-algebra A can be embedded as a subalgebra into Orth(A). We write A ⊆ Orth(A) after identifying A with its image ℎ(A) in Orth(A). According to Theorem 15.5 in [9], the identity 𝐼 is an order unit in Orth(A). By Proposition 3.44, a vector lattice with order unit coincides with the ideal, generated by all selfmajorizing elements. Thus we get Φ3 (Orth(A)) = Orth(A) ⊇ A. It follows for two arbitrary elements 𝑎, 𝜑 ∈ A+ that 𝑎 ∧ 𝑛𝜑 ≤ 𝑐𝑎 𝜑, i. e., all positive elements in A are selfmajorizing. Therefore, each element 𝑎 ∈ A is selfmajorizing as any 𝑎 ∈ A can be represented as 𝑎 = 𝑎+ − 𝑎− , where 𝑎+ , 𝑎− ∈ A+ . Thus we get Φ1 (A) = Φ2 (A) = Φ3 (A) = A.

3.5.4 Finite elements in product algebras Let A be an ℓ-algebra and 𝑝 ∈ ℕ≥2 . The following construction is well-known. For details the reader is referred to [22, 26, 27, 84]. By Π𝑝 (A) := {𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 : 𝑔𝑖 ∈ A for 𝑖 = 1, . . . , 𝑝} ⊆ A we denote the set of all 𝑝-fold products in A. Clearly, Π𝑝 (A) ⊆ A. In general, this inclusion is proper, e. g., if A is as in Example 3.63. Even if A is a semiprime uniformly complete 𝑓-algebra, then in general still Π𝑝 (A) ≠ A; see e. g., [23, p. 136], where an example of a semiprime uniformly complete and not square-root closed 𝑓-algebra is provided. If the set Π𝑝 (A), equipped with the order and algebraic operations induced from A, turns out to be an algebra, then it is called the product algebra of order 𝑝 of A. Denote by Σ𝑝 (A) := {𝑔𝑝 : 𝑔 ∈ A+ } the set of all 𝑝-fold powers of positive elements of A.

64 | 3 Finite, totally finite and selfmajorizing elements

–

For completeness we provide some important properties of Π𝑝 (A) without proofs. If Π𝑝 (A) is a vector space, it may fail to be a vector lattice in general. In case of 𝑝 = 2, there is a counterexample of an ℓ-algebra A, which shows that the vector space Π2 (A) is not a vector lattice under the order induced from A (see [26, Example 1]).

Let A be a uniformly complete 𝑓-algebra and 𝑝 ∈ ℕ≥2 . – The set Π𝑝 (A) is a semiprime uniformly complete 𝑓-subalgebra of A (see [27, Corollary 5.3 (iv)], [26, Corollary 3], and [28, Corollary 4]). – The set Π𝑝 (A) is a vector lattice under the ordering inherited from A, where Π+𝑝 (A) = Σ𝑝(A).

(3.18)

Additionally, for the supremum ∨𝑝 and infimum ∧𝑝 in Π𝑝 (A), the following formulas hold: 𝑓𝑝 ∧𝑝 𝑔𝑝 = (𝑓 ∧ 𝑔)𝑝 –

and

𝑓𝑝 ∨𝑝 𝑔𝑝 = (𝑓 ∨ 𝑔)𝑝

for

𝑓, 𝑔 ∈ A+

(3.19)

If 𝑓1 , . . . , 𝑓𝑝 ∈ A are arbitrary elements, then for the modulus of the product 𝑓1 ⋅ ⋅ ⋅ 𝑓𝑝 in Π𝑝 (A), the following formula is true |𝑓1 ⋅ ⋅ ⋅ 𝑓𝑝 |𝑝 = |𝑓1 | ⋅ ⋅ ⋅ |𝑓𝑝 |.

(3.20)

(see [22],Proposition 1, and [27, Corollary 5.3 (i) and (iv)]). Altogether we obtain for a uniformly complete 𝑓-algebra A and 𝑝 ∈ ℕ≥2 , that Π𝑝 (A) is a semiprime uniformly complete 𝑓-subalgebra of A, where Formulas (3.15), (3.18), (3.19) and (3.20) hold. We now study the finite elements in Π𝑝 (A). Theorem 3.73. Let A be an (Archimedean) uniformly complete 𝑓-algebra, and let 𝑝 ∈ ℕ≥2 . Then 𝑔 ∈ Φ1 (A) with a majorant 𝑢 󳨐⇒ 𝑔𝑝 ∈ Φ1 (Π𝑝 (A)) with the majorant 𝑢𝑝 . If, in addition, A is semiprime, then 𝑔 ∈ Φ1 (A) with a majorant 𝑢 ⇐⇒ 𝑔𝑝 ∈ Φ1 (Π𝑝 (A)) with the majorant 𝑢𝑝 . Proof. ⇒: Without loss of generality we assume 0 < 𝑔 ∈ Φ1 (A). Otherwise consider |𝑔| and apply (d1 ). If 𝑢 ∈ A+ is a majorant of 𝑔, then for each 𝑓 ∈ A+ there is a constant 𝑐𝑓 ≥ 0 with 𝑓 ∧ 𝑛𝑔 ≤ 𝑐𝑓 𝑢 for all 𝑛 ∈ ℕ. Then by means of Formula (3.19) for the 𝑝-th power of 𝑓 we get 𝑓𝑝 ∧𝑝 𝑛𝑝 𝑔𝑝 = 𝑓𝑝 ∧𝑝 (𝑛𝑔)𝑝 = (𝑓 ∧ 𝑛𝑔)𝑝 ≤ (𝑐𝑓𝑢)𝑝 = 𝑐𝑓 𝑝 𝑢𝑝 ,

(3.21)

where the last inequality holds due to the condition (ℓ1 ), (see the footnote on p. 59).

3.5 Finite elements in ℓ-algebras and in product algebras

|

65

Let 𝑓 = 𝑓1 ⋅ ⋅ ⋅ 𝑓𝑝 ∈ Π+𝑝 (A) now be an arbitrary element. By (3.18) we have Π+𝑝 (A) = Σ𝑝(A). Therefore there exists an ℎ ∈ A+ with 𝑓 = ℎ𝑝. By means of (3.21) we get ℎ𝑝 ∧𝑝 𝑛𝑝 𝑔𝑝 ≤ 𝑐ℎ𝑝 𝑢𝑝

for all 𝑛 ∈ ℕ

and (𝑓1 ⋅ ⋅ ⋅ 𝑓𝑝 ) ∧𝑝 𝑛𝑝 𝑔𝑝 = ℎ𝑝 ∧𝑝 𝑛𝑝 𝑔𝑝 ≤ 𝑐ℎ𝑝 𝑢𝑝 𝑝

for all 𝑛 ∈ ℕ.

𝑝

This shows that 𝑔 ∈ Φ1 (Π𝑝 (A)) with the majorant 𝑢 . ⇐: Let 𝑔𝑝 be a positive finite element in Π𝑝 (A). The elements 𝑢1 , . . . , 𝑢𝑝 ∈ A+ exist, such that for arbitrary 𝑎1 , . . . , 𝑎𝑝 ∈ A+ the inequality (𝑎1 ⋅ ⋅ ⋅ 𝑎𝑝 ) ∧𝑝 𝑛𝑔𝑝 ≤ 𝑐𝑎1 ⋅⋅⋅𝑎𝑝 (𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 )

(3.22) 1

holds for all 𝑛 ∈ ℕ and some number 0 < 𝑐𝑎1 ⋅⋅⋅𝑎𝑝 . Since in A the element 𝑢 = (𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 ) 𝑝 exists, the inequality (3.22) can be rewritten as (𝑎1 ⋅ ⋅ ⋅ 𝑎𝑝 ) ∧𝑝 𝑛𝑔𝑝 ≤ 𝑐𝑎1 ⋅⋅⋅𝑎𝑝 𝑢𝑝 .

(3.23)

Now let 𝑎 ∈ A+ . By taking the relation (3.19) and the last inequality into consideration we get 𝑝 𝑛 𝑔)𝑝 = 𝑎𝑝 ∧𝑝 𝑛𝑔𝑝 ≤ 𝑐𝑎𝑝 𝑢𝑝 (𝑎 ∧ √𝑝 𝑛 𝑔)𝑝 = 𝑎𝑝 ∧𝑝 ( √

for all 𝑛 ∈ ℕ.

Due to the semiprimitivity of A, the root is monotone and there holds the inequality 1

𝑝 ((𝑎 ∧ √𝑝 𝑛 𝑔)𝑝) 𝑝 ≤ √𝑐 𝑎𝑝 𝑢. 𝑝 𝑔 ≤ 𝑐𝑎̃ 𝑢 with 𝑐𝑎̃ = √𝑝 𝑐𝑎𝑝 . Therefore, for all 𝑛 ∈ ℕ, there follows the inequality²⁴ 𝑎 ∧ √𝑛 This shows that 𝑔 ∈ Φ1 (A) with 𝑢 as one of its majorants.

Corollary 3.74. Let A be a uniformly complete 𝑓-algebra and let 𝑝 ∈ ℕ≥2 . Then (1) 𝑔1 , . . . , 𝑔𝑝 are finite in A 󳨐⇒ 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 is finite in Π𝑝 (A). If, in addition, A is semiprime, then 1 (2) 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 is finite in Π𝑝 (A) 󳨐⇒ (𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 ) 𝑝 is finite in A; 1

(3) 𝑔1 , . . . , 𝑔𝑝 are finite in A 󳨐⇒ (𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 ) 𝑝 is finite in A; (4) Φ1 (Π𝑝 (A)) ⊆ Φ1 (A); (5) Φ1 (Π𝑝 (A)) = Φ1 (A) ∩ Π𝑝 (A), provided A has the weak factorization property. Proof. (1) Let 𝑔1 , . . . , 𝑔𝑝 be positive finite elements in A. Then the element 𝑔 = 𝑔1 ∨. . .∨ 𝑔𝑝 is also finite in A and, by the previous theorem, the element 𝑔𝑝 is finite in Π𝑝 (𝐴). Since 0 ≤ 𝑔𝑖 ≤ 𝑔 for all 𝑖 = 1, . . . , 𝑝, by the condition (ℓ1 ) we have 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 ≤ 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝−1 𝑔 ≤ 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝−2 𝑔2 ≤ . . . ≤ 𝑔1 𝑔𝑝−1 ≤ 𝑔𝑝 .

𝑝 𝑝 24 For each 𝑚 ∈ ℕ some 𝑛 ∈ ℕ exists such that 𝑚 < √𝑛, so that 𝑎 ∧ 𝑚𝑔 ≤ 𝑎 ∧ √𝑛𝑔 ≤ 𝑐𝑎̃ 𝑢 for all 𝑚 ∈ ℕ.

66 | 3 Finite, totally finite and selfmajorizing elements The element 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 is finite in Π𝑝 (A), since Φ1 (Π𝑝 (A)) is an order ideal in Π𝑝 (A). Let 𝑔1 , . . . , 𝑔𝑝 be arbitrary finite elements in A. By the first part of the proof the element |𝑔1 | ⋅ ⋅ ⋅ |𝑔𝑝 | is finite in Π𝑝 (A). Due to (3.20), we have |𝑔1 | ⋅ ⋅ ⋅ |𝑔𝑝 | = |𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 |𝑝 , and so 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 is a finite element in Π𝑝 (A). Without loss of generality for the proofs of (2) and (3), we may assume that 𝑔1 , . . . , 𝑔𝑝 are positive elements in A, otherwise consider |𝑔1 |, . . . , |𝑔𝑝 | and apply (3.20). 1

According to Remark 3.65 (1) in both cases an element 𝑔 ̃ = (𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝) 𝑝 exists in A. 1

(2) The equality 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 = ((𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 ) 𝑝 )𝑝 = 𝑔̃𝑝 shows that the finiteness of 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 in Π𝑝 (A) implies that 𝑔𝑝̃ is finite. By the theorem one has 𝑔̃ ∈ Φ1 (A). (3) Follows from (1) and (2). Indeed, if 𝑔1 , . . . , 𝑔𝑝 are finite elements in A, then by part (1) the element 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 is finite in Π𝑝 (A), and by part (2) the element 𝑔̃ := 1

(𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝) 𝑝 is finite in A.

1

(4) Let 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 ∈ Φ1 (Π𝑝 (A)). Then by part (2) we get (𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 ) 𝑝 ∈ Φ1 (A), which according to Theorem 3.73 yields 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 ∈ Φ1 (A). (5) The relation “⊆” follows from (4). For the converse relation “⊇”, let 𝜑 ∈ Φ1 (A) ∩ Π𝑝 (A). Then the element 𝜑 can be written as a 𝑝-fold product 𝜑 = 𝜑1 . . . 𝜑𝑝 , and there1

1

fore possesses the root 𝜑 𝑝 ∈ A. By Theorem 3.67 we have 𝜑 𝑝 ∈ Φ1 (A), and by means of (1), then 𝜑 ∈ Φ1 (Π𝑝 (A)). In the next Corollary we obtain some information on totally finite and selfmajorizing elements in an 𝑓-algebra. For its proof we need the following relations. Proposition 3.75. Let A be a uniformly complete 𝑓-algebra and let 𝑝 ∈ ℕ≥2 . Then for all 𝑔 ∈ A, the following implication holds: 𝑔 ∈ 𝑆(A)

󳨐⇒

𝑔𝑝 ∈ 𝑆(Π𝑝 (A)).

(3.24)

⇐⇒

𝑔𝑝 ∈ 𝑆(Π𝑝 (A)).

(3.25)

If, in addition, A is semiprime, then 𝑔 ∈ 𝑆(A)

Proof. Let 𝑔 be a selfmajorizing element in A, i. e., |𝑔| is a majorant of 𝑔 in A. By Theorem 3.73 this implies that |𝑔|𝑝 is a majorant of 𝑔𝑝 in Π𝑝 (A). Formula (3.20) yields the equality |𝑔|𝑝 = |𝑔𝑝 |𝑝 , so |𝑔𝑝 |𝑝 is a majorant of 𝑔𝑝 in Π𝑝 (A). Therefore 𝑔𝑝 ∈ 𝑆(Π𝑝 (A)). Conversely, let 𝑔𝑝 ∈ 𝑆(Π𝑝 (A)), i. e., |𝑔𝑝 |𝑝 is a majorant of 𝑔𝑝 in Π𝑝 (A). The equality |𝑔𝑝 |𝑝 = |𝑔|𝑝 and Theorem 3.73 imply that |𝑔| is a majorant of 𝑔 in A, and therefore 𝑔 ∈ 𝑆(A). Corollary 3.76. Let A be a uniformly complete 𝑓-algebra, and let 𝑝 ∈ ℕ≥2 . Then the following implications hold for all 𝑔 ∈ A: (1) 𝑔 ∈ Φ2 (A) 󳨐⇒ 𝑔𝑝 ∈ Φ2 (Π𝑝 (A)); (2) 𝑔 ∈ Φ3 (A) 󳨐⇒ 𝑔𝑝 ∈ Φ3 (Π𝑝 (A)). If, in addition, A is semiprime, then the converse implications are also true.

3.5 Finite elements in ℓ-algebras and in product algebras

| 67

Proof. (1). ⇒: Let 𝑔 ∈ Φ2 (A) have a finite majorant 𝑢 ∈ A. By the first part of Theorem 3.73 we obtain 𝑔𝑝 ∈ Φ1 (Π𝑝 (A)), with majorant 𝑢𝑝 , and the same theorem guarantees the finiteness of the majorant 𝑢𝑝 in Π𝑝 (A), i. e., 𝑔𝑝 ∈ Φ2 (Π𝑝 (A)). ⇐: Let 𝑔𝑝 ∈ Φ2 (Π𝑝 (A)) with a finite majorant 𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 . By Remark 3.65 (1) we can 1

𝑝

write this majorant as a 𝑝-fold product 𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 = ((𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 ) 𝑝 ) = 𝑢𝑝 of the element 1 𝑝

𝑢 = (𝑢1 ⋅ ⋅ ⋅ 𝑢𝑝 ) . Then the semiprimitivity of A and Theorem 3.73 yield 𝑔 ∈ Φ1 (A) with the majorant 𝑢, and also the finiteness of the majorant 𝑢 in A. Therefore 𝑔 ∈ Φ2 (A). (2). The set Φ3 (A) coincides with the ideal generated by the set 𝑆(A) (see Corollary 3.40), i. e., 𝑛

󵄨 󵄨 Φ3 (A) = {𝑎 ∈ A : ∃𝑠1 , . . . , 𝑠𝑛 ∈ 𝑆(A) and 𝜆 1 , . . . , 𝜆 𝑛 ∈ ℝ+ with |𝑎| ≤ ∑ 𝜆 𝑖 󵄨󵄨󵄨𝑠𝑖 󵄨󵄨󵄨 }. 𝑖=1

󵄨 󵄨 Since ∑𝑛𝑖=1 𝜆 𝑖 󵄨󵄨󵄨𝑠𝑖 󵄨󵄨󵄨 is a positive selfmajorizing element (see Proposition 3.36), the order ideal Φ3 (A) can be written as Φ3 (A) = {𝑎 ∈ A : ∃𝑠 ∈ 𝑆+ (A) : |𝑎| ≤ 𝑠}. ⇒: Let 𝑔 ∈ Φ3 (A). There is an 𝑠 ∈ 𝑆+(A) such that |𝑔| ≤ 𝑠. Since 𝑠 is a majorant of 𝑠 in A, the element 𝑠 is also a majorant of |𝑔|. By the first part of Theorem 3.73 we obtain that the element |𝑔|𝑝 is finite in Π𝑝 (A) with a majorant 𝑠𝑝 . Due to (3.24), the element 𝑠𝑝 is selfmajorizing in Π𝑝 (A). Formula (3.20) yields that the element |𝑔𝑝 |𝑝 belongs to the ideal generated by 𝑆+ (Π𝑝 (A)), i. e., 𝑔𝑝 ∈ Φ3 (Π𝑝 (A)). ⇐: Conversely, let 𝑔𝑝 ∈ Φ3 (Π𝑝 (A)). Since Φ3 (Π𝑝 (A)) is the ideal generated by 𝑆+(Π𝑝 (A)) in Π𝑝 (A), there is an element 𝑠 ∈ 𝑆+ (Π𝑝 (A)) such that |𝑔𝑝 |𝑝 ≤ 𝑠. Using 1

Remark 3.65 (1) we can write the majorant 𝑠 as 𝑠 = 𝑠1 ⋅ ⋅ ⋅ 𝑠𝑝 = 𝑠 ̃ 𝑝 , where 𝑠 ̃ := (𝑠1 ⋅ ⋅ ⋅ 𝑠𝑝 ) 𝑝 . Notice that 𝑠 ̃ 𝑝 has itself as a majorant in Π𝑝 (A). Due to (3.25), and the second part of Theorem 3.73, the element 𝑠 ̃ is selfmajorizing in A and is a majorant of 𝑔 in A. Therefore we obtain 𝑔 ∈ Φ3 (A). By summing up the results obtained in Theorem 3.73 and Corollaries 3.74 and 3.76 we formulate Corollary 3.77. Let A be a semiprime uniformly complete 𝑓-algebra and 𝑝 ∈ ℕ≥2 . Then for 𝑖 = 1, 2, 3 there holds 𝑝 (Φ𝑖 (A)) = Φ𝑖 (Π𝑝 (A)), 𝑝

where (Φ𝑖 (A)) = {𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 ∈ Π𝑝 (A) : 𝑔1 , . . . , 𝑔𝑝 ∈ Φ𝑖 (A)}. Proof. Let 𝑖 = 1. Indeed, the relation “⊆” follows from Corollary 3.74 (1). The relation “⊇” is obtained as follows: Let 𝑔 ∈ Φ1 (Π𝑝 (A)), i. e., 𝑔 = 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 with 𝑔𝑖 ∈ A. Then 1

1

Corollary 3.74 (2) implies that 𝑔 𝑝 is a finite element in A. From 𝑔 = (𝑔 𝑝 )𝑝 it is clear that 𝑝 𝑔 is a product consisting of 𝑝 finite elements of A, i. e., 𝑔 ∈ (Φ1 (A)) . The cases 𝑖 = 2, 3 are proved similarly by using Corollary 3.76. The proof of the second inclusion of the previous corollary (for 𝑖 = 1), shows that each finite element of Π𝑝 (A) has a representation as the 𝑝-th power of a single finite

68 | 3 Finite, totally finite and selfmajorizing elements element of A. In general, 𝑔 = 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 ∈ Φ1 (Π𝑝 (A)) does not imply 𝑔1 , . . . , 𝑔𝑝 ∈ Φ1 (A), as demonstrated by the next example. Example 3.78. Example of an Archimedean semiprime uniformly complete 𝑓-algebra A, such that 𝑔1 ⋅ ⋅ ⋅ 𝑔𝑝 ∈ Φ1 (Π𝑝 (A)) does not imply 𝑔1 , . . . , 𝑔𝑝 ∈ Φ1 (A). Let A = 𝐶([0, ∞)) be the vector lattice of all continuous functions on the interval [0, ∞), equipped with pointwise order and algebraic operations. Then A is an Archime-

dean unitary semiprime uniformly complete 𝑓-algebra. For 𝑝 = 3 consider Π3 (A) = {𝑓1 𝑓2 𝑓3 : 𝑓1 , 𝑓2 , 𝑓3 ∈ A} . Since the function 1[0,∞) is the multiplicative unit in A, all functions of A belong to Π3 (A). This means that A and Π3 (A) coincide. The finite elements in A are exactly the functions with compact support. Consider the following three functions of A: 𝑓2 (𝑡) = 1[0,∞);

𝑓1 (𝑡) = 𝑡;

and

𝑓3 (𝑡) = {

sin 𝑡, 0,

𝑡 ∈ [0, 𝜋), 𝑡 ∈ [𝜋, ∞).

The only finite element among them is 𝑓3 . The product 𝑓1 𝑓2 𝑓3 , i. e., the function 𝜑(𝑡) = {

for 𝑡 ∈ [0, 𝜋), for 𝑡 ∈ [𝜋, ∞)

𝑡 sin 𝑡, 0,

is a finite element in A = Π3 (A), however not all of its factors are finite elements. From Corollary 3.77 we know that a finite function 𝑓 ̃ exists in A such that 𝑓 ̃ 3 = 𝑓1 𝑓2 𝑓3 . In our case this is the function 1

3 ̃ = { (𝑡 sin 𝑡) , 𝑓(𝑡) 0,

𝑡 ∈ [0, 𝜋), 𝑡 ∈ [𝜋, ∞) .

Figure 3.3 illustrates the functions 𝜑 = 𝑓1 𝑓2 𝑓3 and 𝑓 ̃ = √3 𝜑:

𝑓1

𝜑 𝑓2 𝑓3

𝑓̃

𝜋

Fig. 3.3. The functions 𝜑 and 𝑓 ̃

4 Finite elements in vector lattices of linear operators The famous Riesz–Kantorovich Theorem (Theorem 2.4) states that for an arbitrary vector lattice 𝐸, and an arbitrary Dedekind complete vector lattice 𝐹, the ordered vector space L 𝑏 (𝐸, 𝐹) of all order bounded operators is a vector lattice (even Dedekind complete), and coincides with the space of all regular operators. In that case, according to Theorem 3.13, one has Φ1 (L 𝑏 (𝐸, 𝐹)) = Φ2 (L 𝑏 (𝐸, 𝐹)). These facts are the starting point for a further investigation of finite elements in vector lattices of operators. There are many other classes of linear operators which turn out to be vector lattices or, if equipped with an appropriate norm, even Banach lattices. The results of the present section were obtained mainly in the papers [38] and [54]. If L 𝑏 (𝐸, 𝐹) is a vector lattice, then obviously each operator 𝑇 ∈ L 𝑏 (𝐸, 𝐹) can be represented as 𝑇 = 𝑇+ − 𝑇− , with 𝑇+ , 𝑇− ∈ L+ (𝐸, 𝐹), and possesses its modulus |𝑇| in L 𝑏 (𝐸, 𝐹). As a consequence, it follows that L 𝑏 (𝐸, 𝐹) coincides with the vector space generated by the positive operators, and therefore L 𝑟 (𝐸, 𝐹) = L 𝑏 (𝐸, 𝐹). Then due to Theorem 2.5 the vector lattice 𝐹 is Dedekind complete. For normed vector lattices 𝐸 and 𝐹, the space L(𝐸, 𝐹) of all linear continuous operators from 𝐸 to 𝐹 is equipped with the standard (uniform) operator norm ‖𝑇‖ = sup{‖𝑇𝑥‖ : 𝑥 ∈ 𝐸, ‖𝑥‖ ≤ 1}.

(4.1)

As already mentioned (see p. 15), L+ (𝐸, 𝐹) ⊂ L 𝑟 (𝐸, 𝐹) ⊂ L(𝐸, 𝐹). In this chapter we investigate all kinds of finite elements in several vector lattices of operators, i. e., for two (normed) vector lattices 𝐸 and 𝐹 our investigation will be done in a given vector lattice V(𝐸, 𝐹) of (continuous) linear operators. For an operator 𝑇 : 𝐸 → 𝐹, its adjoint operator 𝑇󸀠 : 𝐹󸀠 → 𝐸󸀠 is defined by (𝑇󸀠 𝑓)𝑥 = 𝑓(𝑇𝑥) for any 𝑓 ∈ 𝐹󸀠 , and 𝑥 ∈ 𝐸. Then ‖𝑇‖ = ‖𝑇󸀠 ‖ and 0 ≤ 𝑇 implies 0 ≤ 𝑇󸀠 . With respect to the operator norm, the space L 𝑟 (𝐸, 𝐹) is not complete (see for example [1]). However, a natural norm exists on L 𝑟 (𝐸, 𝐹), the regular norm ‖ ⋅ ‖𝑟 , defined by ‖𝑇‖𝑟 = inf { ‖𝑆‖ : 𝑆 ∈ 𝐿 + (𝐸, 𝐹), ±𝑇 ≤ 𝑆 } for 𝑇 ∈ L 𝑟 (𝐸, 𝐹), (4.2) which makes L 𝑟 (𝐸, 𝐹) a Banach space. If 𝐹 is Dedekind complete, then (L 𝑟 (𝐸, 𝐹), ‖⋅‖𝑟 ) is a Banach lattice, and ‖𝑇‖𝑟 = ‖|𝑇|‖ for all 𝑇 ∈ L 𝑟 (𝐸, 𝐹) (see [95, Proposition 1.3.6]). In particular, in this case, each operator 𝑇 ∈ L 𝑟 (𝐸, 𝐹) possesses its modulus |𝑇|. In Section 4.1 we show, in particular, that every lattice isomorphism from 𝐸 to 𝐹 is a selfmajorizing element in L 𝑟 (𝐸, 𝐹).

70 | 4 Finite elements in vector lattices of linear operators In Section 4.2 it is shown that if 𝐸 is an 𝐴𝐿-space and 𝐹 is a Dedekind complete 𝐴𝑀-space with an order unit, then each regular operator is a finite element in L 𝑟 (𝐸, 𝐹). In Section 4.3 the finiteness of finite rank operators in L 𝑟 (𝐸, 𝐹) is dealt with in the case that 𝐸 and 𝐹 are Banach lattices. A necessary and sufficient condition is given for a rank one operator to be a finite element in the vector lattice L 𝑟 (𝐸, 𝐹). Vector lattices of other operators, e. g., compact, weakly compact, Dunford–Pettis and others will be considered in Section 4.4. In Section 4.5, for some vector lattices of operators conditions are found under which they coincide with the ideal of all finite elements. In Section 4.6 it is shown that those rank one and finite rank operators which are assembled by means of finite elements from 𝐸󸀠 and 𝐹 are finite elements in V(𝐸, 𝐹). Finally, in Section 4.7 the question of how the order structures of 𝐸, 𝐹, and V(𝐸, 𝐹) are mutually related will be considered.

4.1 Some general results Let us first sum up some results we have obtained so far for orthomorphisms. From Corollary 3.60 (1), we know that for a vector lattice 𝐸, the sets Φ𝑖 (Orth(𝐸)), 𝑖 = 1, 2 are both an 𝑓-subalgebra and a ring ideal in Orth(𝐸). If 𝐸 is a Banach lattice or a unitary 𝑓-algebra which admits a submultiplicative Riesz norm, then Orth(𝐸) possesses an order unit and coincides with Φ𝑖 (Orth(𝐸)) for 𝑖 = 1, 2, 3, (Corollaries 3.58, and 3.60 (2)). According to Theorem 3.72 for a semiprime 𝑓-algebra 𝐸, which is also a Banach lattice, one has Orth(𝐸) = Φ𝑖 (Orth(𝐸)) for 𝑖 = 1, 2, 3. Now we show that for a Dedekind complete Banach lattice 𝐸 the orthomorphisms are totally finite elements in the vector lattice L 𝑟 (𝐸). We then extend this result to lattice isomorphisms, study the behaviour of finiteness between an operator and its adjoint, and obtain some results under the natural embeddings of the Banach lattices. Finally, we show that the finite elements and their majorants in L 𝑟 (𝐸, 𝐹󸀠󸀠 ) and L 𝑟 (𝐹󸀠 , 𝐸󸀠 ) are in a one-to-one correspondence, where 𝐸󸀠 , 𝐹󸀠 are the Banach duals of 𝐸, 𝐹, and 𝐹󸀠󸀠 the second dual of 𝐹. Remember that an operator 𝑇 : 𝐸 → 𝐸 is an orthomorphism if 𝑇 ∈ L 𝑏 (𝐸), and 𝑇 is band preserving in 𝐸. Although the next result is a special case of Theorem 4.2, we prove it separately in order to demonstrate a direct application of Theorem 3.42. Theorem 4.1. Let 𝐸 be a Dedekind complete Banach lattice. Then Orth(𝐸) ⊂ Φ1 (L 𝑟 (𝐸)) = Φ2 (L 𝑟 (𝐸)). In particular, the identity operator 𝐼 on 𝐸 is a selfmajorizing element in L 𝑟 (𝐸).

4.1 Some general results

|

71

Proof. The Dedekind completeness of 𝐸 implies L 𝑟 (𝐸) = L 𝑏 (𝐸), and that L 𝑟 (𝐸) is a Dedekind complete vector lattice. Therefore, due to Theorem 3.13, Φ1 (L 𝑟 (𝐸)) = Φ2 (L 𝑟 (𝐸)). By the celebrated Wickstead’s result ([9, Theorem 15.5]) for the Banach lattice 𝐸 it holds that Orth(𝐸) = {𝑇 ∈ L 𝑏 (𝐸) : ∃ 𝜆 > 0, −𝜆𝐼 ≤ 𝑇 ≤ 𝜆𝐼}, and Orth(𝐸) equipped with the corresponding order unit norm ‖𝑇‖𝐼 = inf{𝜆 > 0 : |𝑇| ≤ 𝜆𝐼} is an 𝐴𝑀-space with 𝐼 as an order unit. In particular, Orth(𝐸) is equal to the ideal generated by 𝐼 in L 𝑟 (𝐸). According to Theorem 8.11 of [9], the space Orth(𝐸) coincides with the band {𝐼}⊥⊥ generated by the identity operator 𝐼 in L 𝑟 (𝐸). Now the result follows from Theorems 3.11 and 3.9. Theorem 3.42 implies that 𝐼 is a selfmajorizing element, since any band in L 𝑟 (𝐸) is a projection band. The same result can be obtained by avoiding the Dedekind completeness of 𝐸 if L 𝑟 (𝐸) is only known to be a vector lattice¹; see Theorem 4.32 below. That the identity operator 𝐼𝐸 on a Dedekind complete Banach lattice 𝐸 is a selfmajorizing element in L 𝑟 (𝐸) (the last statement of the theorem) means that for any 𝑈 ∈ L 𝑟 (𝐸) a positive number 𝑐𝑈 exists such that |𝑈| ∧ 𝑛 𝐼𝐸 ≤ 𝑐𝑈 𝐼𝐸

for all 𝑛 ∈ ℕ .

(4.3)

This result can be extended to lattice isomorphisms, i. e., to bijective and bipositive operators. For the proof of this result we use the fact that for an order continuous² lattice homomorphism 𝑇 : 𝐸 → 𝐹, where 𝐸 and 𝐹 are vector lattices with 𝐹 Dedekind complete, and any 𝑈, 𝑈1 , 𝑈2 ∈ L 𝑟 (𝐸), the following equalities |𝑇𝑈| = 𝑇|𝑈|,

𝑇(𝑈1 ∧ 𝑈2 ) = (𝑇𝑈1 ) ∧ (𝑇𝑈2 )

(4.4)

hold. Indeed, by means of Theorem 1.16 of [9], for each 𝑥 ∈ 𝐸+ one has 𝑛

𝑛

{ ∑ |𝑈𝑥𝑖 | : ∀𝑥𝑖 ∈ 𝐸+ with ∑ 𝑥𝑖 = 𝑥, 𝑛 ∈ ℕ} ↑ |𝑈|𝑥 , 𝑖=1

𝑖=1

from where it follows that 𝑛

𝑛

|𝑇𝑈|𝑥 = sup { ∑ |𝑇𝑈𝑥𝑖 | : ∀𝑥𝑖 ∈ 𝐸+ with ∑ 𝑥𝑖 = 𝑥, 𝑛 ∈ ℕ} 𝑖=1

𝑖=1 𝑛

𝑛

= 𝑇 ( sup { ∑ |𝑈𝑥𝑖 | : ∀𝑥𝑖 ∈ 𝐸+ with ∑ 𝑥𝑖 = 𝑥, 𝑛 ∈ ℕ}) 𝑖=1

𝑖=1

= 𝑇|𝑈|𝑥

1 And L 𝑟 (𝐸) ⫋ L 𝑏 (𝐸), otherwise, according to Theorem 2.5 mentioned on p. 10, 𝐸 is Dedekind complete. 2 An operator 𝑇 : 𝐸 → 𝐹 is called order continuous if 0 ≤ 𝑥𝛼 ↑ 𝑥 in 𝐸 implies 𝑇𝑥𝛼 ↑ 𝑇𝑥 in 𝐹; see Definition 2.1.

72 | 4 Finite elements in vector lattices of linear operators for all 𝑥 ∈ 𝐸+ . Therefore, the first equality is proved. The second one is similarly proved. Theorem 4.2. Let 𝐸 be a Banach lattice which is lattice isomorphic to a Dedekind complete vector lattice 𝐹. Then each surjective lattice isomorphism 𝑇 : 𝐸 → 𝐹 is a selfmajorizing element in the vector lattice L 𝑟 (𝐸, 𝐹). Proof. Since 𝑇 is a lattice isomorphism, 𝐸 is Dedekind complete as well. If 𝑆 ∈ L 𝑟 (𝐸, 𝐹), then obviously 𝑇−1 𝑆 ∈ L 𝑟 (𝐸), and therefore (4.3) implies |𝑇−1 𝑆| ∧ 𝑛 𝐼𝐸 ≤ 𝑐𝑇−1 𝑆 𝐼𝐸

for all 𝑛 ∈ ℕ,

(4.5)

as any lattice isomorphism the operator 𝑇 is order continuous. By applying the (positive) operator 𝑇 to the last inequality, by means of (4.4), it follows that 𝑇 (|𝑇−1 𝑆| ∧ 𝑛 𝐼𝐸 ) = |𝑆| ∧ 𝑛 𝑇 ≤ 𝑐𝑇−1𝑆 𝑇

for all 𝑛 ∈ ℕ ,

which by definition shows that the operator 𝑇 is a selfmajorizing element in L 𝑟 (𝐸, 𝐹). Remark 4.3. If 𝑇 is a lattice isomorphism from 𝐸 only onto 𝑇(𝐸) ⊂ 𝐹, where 𝑇(𝐸) ≠ 𝐹, then 𝑇 fails to be even a finite element in L 𝑟 (𝐸, 𝐹) in general. The corresponding Example 4.21 is conveniently provided after Corollary 4.20. In the next theorem we return to the question concerning the relations between the finite elements in a vector lattice of operators and the finite ones in a special projection band (cf. Section 3.3.1) of the latter. Theorem 4.4. Let 𝐸 and 𝐹 be Banach lattices such that 𝐹 is Dedekind complete. Let 𝐻 be a band³ of 𝐹, and 𝑃 : 𝐹 → 𝐻 the band projection. Then (1) L 𝑟 (𝐸, 𝐻) is a projection band of L 𝑟 (𝐸, 𝐹); (2) Φ1 (L 𝑟 (𝐸, 𝐻)) = Φ1 (L 𝑟 (𝐸, 𝐹)) ∩ L 𝑟 (𝐸, 𝐻) = {𝑃𝑇: 𝑇 ∈ Φ1 (L 𝑟 (𝐸, 𝐹))}. Proof. It is obvious that L 𝑟 (𝐸, 𝐻) is an ideal of L 𝑟 (𝐸, 𝐹). If 𝑇𝛼 ∈ L 𝑟 (𝐸, 𝐻) is such that 𝑇𝛼 ↑ 𝑇 in L 𝑟 (𝐸, 𝐹), then it is clear that 𝑃𝑇𝛼 = 𝑇𝛼 ↑ 𝑃𝑇 in L 𝑟 (𝐸, 𝐹). It is 𝑇 = 𝑃𝑇 since the order limit is uniquely defined, and thus 𝑇 ∈ L 𝑟 (𝐸, 𝐻), i. e., L 𝑟 (𝐸, 𝐻) is a band and, due to the Dedekind completeness of L 𝑟 (𝐸, 𝐹), assertion (1) holds. Let 𝑃∗ : L 𝑟 (𝐸, 𝐹) → L 𝑟 (𝐸, 𝐻) be the band projection. For each 𝑇 ∈ L 𝑟 (𝐸, 𝐹) ⊥ it is easy to see that 𝑃𝑇 ∈ L 𝑟 (𝐸, 𝐻), and (𝐼 − 𝑃)𝑇 ∈ (L 𝑟 (𝐸, 𝐻)) . It follows that ∗ 𝑟 𝑃 (𝑇) = 𝑃𝑇 for all 𝑇 ∈ L (𝐸, 𝐹). Statement (2) is proved by means of Theorem 3.24 and Proposition 3.14 as follows: since L 𝑟 (𝐸, 𝐻) is a band in L 𝑟 (𝐸, 𝐹), Theorem 3.24 implies Φ1 (L 𝑟 (𝐸, 𝐻)) ⊂ Φ1 (L 𝑟 (𝐸, 𝐹)) ∩ L 𝑟 (𝐸, 𝐻). Then the inverse inclusion follows from Proposition 3.14 because 𝑃∗ is a positive projection onto L 𝑟 (𝐸, 𝐻).

3 In a Dedekind complete vector lattice any band is a projection band.

4.1 Some general results |

73

Let 𝐸 and 𝐹 be Banach lattices. Consider the scheme 𝑇

󳨀→

𝐸 𝐸󸀠

𝑇󸀠

󳨀→

𝑗

𝐹󸀠󸀠

←󳨄

𝐹

𝐹󸀠󸀠󸀠

𝐽

𝐹󸀠

←󳨄

where 𝐹󸀠 , 𝐹󸀠󸀠 , and 𝐹󸀠󸀠󸀠 denote the first, second, and third norm dual of 𝐹 respectively,⁴ 𝑗 : 𝐹 󳨅→ 𝐹󸀠󸀠 and 𝐽: 𝐹󸀠 󳨅→ 𝐹󸀠󸀠󸀠 denote the corresponding canonical embeddings⁵ and 𝑇󸀠 : 𝐹󸀠󸀠󸀠 → 𝐸󸀠 is the adjoint operator to 𝑇 : 𝐸 → 𝐹󸀠󸀠 . Recall that all duals of the Banach lattices 𝐸 and 𝐹 are Dedekind complete Banach lattices on their own; see Corollary 2.14. Define the mapping 𝑟

󸀠󸀠

𝑟

󸀠

󸀠

P : L (𝐸, 𝐹 ) → L (𝐹 , 𝐸 )

by P(𝑇) = 𝑇󸀠 𝐽 for 𝑇 ∈ L 𝑟 (𝐸, 𝐹󸀠󸀠 ).

The following fact is established among others in Theorem 5.6 of [40] and will be used further on: (†) P is an order continuous isometric lattice isomorphism from (L 𝑟 (𝐸, 𝐹󸀠󸀠 ), ‖ ⋅ ‖𝑟 ) onto (L 𝑟 (𝐹󸀠 , 𝐸󸀠 ), ‖ ⋅ ‖𝑟 ). Theorem 4.5. Let 𝐸 and 𝐹 be Banach lattices, and P defined as above. Denote A = L 𝑟 (𝐸, 𝐹󸀠󸀠 ) and B = L 𝑟 (𝐹󸀠 , 𝐸󸀠 ). Then (1) 𝑇 ∈ A is finite if and only if P(𝑇) is finite in B, i. e., P (Φ1 (A)) = Φ1 (B); (2) 𝑈 ∈ L+ (𝐸, 𝐹󸀠󸀠 ) is an A-majorant of 𝑇 if and only if P(𝑈) is a B-majorant of P(𝑇); (3) P (Φ𝑖 (A)) = Φ𝑖 (B), i = 1, 2, 3. Proof. For 𝑇 ∈ Φ1 (A) let 𝑈 ∈ L+(𝐸, 𝐹󸀠󸀠 ) be an A-majorant of 𝑇. Then for each 𝑆 ∈ A there is a number 𝑐𝑆 > 0 such that |𝑆| ∧ 𝑛|𝑇| ≤ 𝑐𝑆 𝑈

for all 𝑛 ∈ ℕ .

(4.6)

The fact (†) implies that for any 𝑊 ∈ B there exists 𝑆 ∈ A such that 𝑊 = P(𝑆), and so (†), (4.4), and (4.6) yield |𝑊| ∧ 𝑛|P(𝑇)| = P(|𝑆| ∧ 𝑛|𝑇|) ≤ 𝑐𝑆 P(𝑈) for all 𝑛 ∈ ℕ, which shows that P(𝑇) is a finite element in B, and P(𝑈) is its B-majorant. On the other hand, if 𝑉 ∈ Φ1 (B), and 𝑊 ∈ L+ (𝐹󸀠 , 𝐸󸀠 ) is a B-majorant of 𝑉, then 𝑉0 ∈ A and 𝑊0 ∈ L+ (𝐸, 𝐹󸀠󸀠 ) exist such that 𝑉 = P(𝑉0 ), and 𝑊 = P(𝑊0 ). Now, in a similar way it is easy to verify that 𝑉0 is a finite element in A, and 𝑊0 is an Amajorant of 𝑉0 . Therefore statements (1) and (2) are proved. Since P(𝑈) is a B-majorant for P(𝑇) whenever 𝑈 is an A-majorant for 𝑇, statement (3) immediately follows from (1) and (2).

4 Analogously for 𝐸. 5 Or natural inclusions.

74 | 4 Finite elements in vector lattices of linear operators For simplicity the following corollaries are formulated only for the case of finite elements. From statement (2) of the theorem, however, it is clear that the image under P of any majorant of an operator 𝑇 ∈ A is a majorant of P(𝑇). Therefore the corollaries also hold for totally finite and selfmajorizing elements in the corresponding vector lattices of regular operators. Note that if 𝑇 ∈ L 𝑟 (𝐸, 𝐹), then 𝑗𝑇 ∈ A. An easy calculation now shows that 𝑇󸀠 = P(𝑗𝑇). This leads to the following corollary. Corollary 4.6. Let 𝐸 and 𝐹 be Banach lattices and 𝑇 ∈ L 𝑟 (𝐸, 𝐹). Then 𝑇󸀠 is finite in L 𝑟 (𝐹󸀠 , 𝐸󸀠 ) if and only if 𝑗𝑇 is finite in L 𝑟 (𝐸, 𝐹󸀠󸀠 ). As a special case, when 𝐹 is a reflexive Banach lattice, we obtain that the finiteness of an operator 𝑇 : 𝐸 → 𝐹 can be characterized by the finiteness of its adjoint 𝑇󸀠 . Corollary 4.7. If 𝐹 is a reflexive Banach lattice, then for each Banach lattice 𝐸, an operator 𝑇 ∈ L 𝑟 (𝐸, 𝐹) is a finite element if and only if 𝑇󸀠 is finite in L 𝑟 (𝐹󸀠 , 𝐸󸀠 ), i. e., Φ1 (L 𝑟 (𝐹󸀠 , 𝐸󸀠 )) = {𝑇󸀠 : 𝑇 ∈ Φ1 (L 𝑟 (𝐸, 𝐹))}. Moreover, 𝑈 ∈ L+ (𝐸, 𝐹) is an L 𝑟 (𝐸, 𝐹)-majorant of 𝑇 if and only if 𝑈󸀠 is an L 𝑟 (𝐹󸀠 , 𝐸󸀠 )majorant of 𝑇󸀠 . Since the identity operator 𝐼𝐸󸀠 is a finite element in L 𝑟 (𝐸󸀠 ) (see Theorem 4.1) and P(𝑖) = 𝐼𝐸󸀠 , where 𝑖 : 𝐸 󳨅→ 𝐸󸀠󸀠 is the natural inclusion, the next result is immediate. Corollary 4.8. Let 𝐸 be a Banach lattice. Then the natural inclusion 𝑖 : 𝐸 󳨅→ 𝐸󸀠󸀠 is a finite element in L 𝑟 (𝐸, 𝐸󸀠󸀠 ). By using the facts leading to Formula (4.4), similar for a lattice isomorphism 𝑇 : 𝐸 → 𝐹, and for operators 𝑈, 𝑈1 , 𝑈2 ∈ L 𝑟 (𝐹), one has |𝑈𝑇| = |𝑈|𝑇,

(𝑈1 ∧ 𝑈2 )𝑇 = (𝑈1 𝑇) ∧ (𝑈2 𝑇).

The previous result yields Corollary 4.9. Let 𝐸 and 𝐹 be Banach lattices such that 𝐹 is Dedekind complete. If 𝑇 : 𝐸 → 𝐹 is a lattice isomorphism between 𝐸 and 𝐹, then 𝑗𝑇 is a finite element in L 𝑟 (𝐸, 𝐹󸀠󸀠 ). The operator 𝑗𝑇 is a totally finite or a selfmajorizing element in L 𝑟 (𝐸, 𝐹󸀠󸀠 ) if the inclusion 𝑗 is of such kind in L 𝑟 (𝐹, 𝐹󸀠󸀠 ) respectively. Proof. From the previous corollary we know that 𝑗 : 𝐹 → 𝐹󸀠󸀠 is finite in L 𝑟 (𝐹, 𝐹󸀠󸀠 ). This means there is an operator 𝑈1 ∈ L 𝑟 (𝐹, 𝐹󸀠󸀠 ) which is a majorant of 𝑗. Observe that for 𝑆 ∈ L 𝑟 (𝐸, 𝐹󸀠󸀠 ), the operator |𝑆|𝑇−1 is positive and obviously belongs to L 𝑟 (𝐹, 𝐹󸀠󸀠 ), so that for some positive number 𝑐|𝑆|𝑇−1 one has |𝑆|𝑇−1 ∧ 𝑛𝑗 ≤ 𝑐|𝑆|𝑇−1 𝑈1

for all 𝑛 ∈ ℕ.

Due to 𝑇 ≥ 0, the last inequality implies (|𝑆|𝑇−1 ∧ 𝑛𝑗) 𝑇 ≤ 𝑐|𝑆|𝑇−1 𝑈1 𝑇

for all 𝑛 ∈ ℕ.

4.2 Finiteness of regular operators on 𝐴𝐿-spaces

|

75

By means of Theorem 1.16 of [9] and 𝑇−1 ≥ 0, one has for each 𝑥 ∈ 𝐸+ the relation 𝑘

𝑘

(|𝑆|𝑇−1 ∧ 𝑛 𝑗) (𝑇𝑥) = inf { ∑ |𝑆|𝑇−1 (𝑦𝑖 ) ∧ 𝑛 𝑗(𝑦𝑖 ) : 𝑦𝑖 ∈ 𝐹+ , ∑ 𝑦𝑖 = 𝑇𝑥} 𝑖=1

𝑖=1

𝑘

𝑘

= inf { ∑ |𝑆|(𝑥𝑖 ) ∧ 𝑛 𝑗𝑇(𝑥𝑖 ) : 𝑥𝑖 ∈ 𝐸+ , ∑ 𝑥𝑖 = 𝑥, 𝑇𝑥𝑖 = 𝑦𝑖 } 𝑖=1

𝑖=1

= (|𝑆| ∧ 𝑛 𝑗𝑇) (𝑥) for all 𝑛 ∈ ℕ, which shows⁶ that the operator 𝑗𝑇 is a finite element in L 𝑟 (𝐹, 𝐹󸀠󸀠 ), with 𝑈1 𝑇 as a majorant. If 𝑈1 = 𝑗 (i. e., 𝑗 is supposed to be selfmajorizing), then 𝑗𝑇 is selfmajorizing as well. Consider the case 𝑗 ∈ Φ2 (L 𝑟 (𝐹, 𝐹󸀠󸀠 )). Then 𝑈1 can be assumed to be a finite element in L 𝑟 (𝐹, 𝐹󸀠󸀠 ) with a majorant 𝑉1 ∈ L 𝑟 (𝐹, 𝐹󸀠󸀠 ) and, we have to show that 𝑈1 𝑇 is also a finite element. Indeed, for 𝑆 ∈ L 𝑟 (𝐸, 𝐹󸀠󸀠 ) and the operator |𝑆|𝑇−1 one has |𝑆|𝑇−1 ∧ 𝑛 𝑈1 ≤ 𝑐|𝑆|𝑇−1 𝑉1

for all 𝑛 ∈ ℕ.

A similar calculation as above shows that (|𝑆| 𝑇−1 ∧ 𝑛 𝑈1 )𝑇 = |𝑆| ∧ 𝑛 𝑈1 𝑇. Due to 𝑇 ≥ 0 for the last operator one has −1 |𝑆| ∧ 𝑛 𝑈1 𝑇 = (|𝑆| 𝑇 ∧ 𝑛 𝑈1 )𝑇 ≤ 𝑐|𝑆|𝑇−1 𝑉1 𝑇.

This shows that 𝑈1 𝑇 is a finite element and has 𝑉1 𝑇 as one of its L 𝑟 (𝐸, 𝐹󸀠󸀠 )-majorants.

4.2 Finiteness of regular operators on 𝐴𝐿-spaces Now we turn our attention to finite elements in the vector lattice of regular operators defined on 𝐴𝐿-spaces and address the question when is each regular operator a finite element? A normed vector lattice 𝐸 has a Levi norm or equivalently, is said to satisfy⁷ the con󵄩 󵄩 dition (B), or also, has a monotonically complete norm, if 0 ≤ 𝑥𝛼 ↑, 𝑥𝛼 ∈ 𝐸, sup 󵄩󵄩󵄩𝑥𝛼 󵄩󵄩󵄩 < ∞ imply the existence of an element 𝑥 ∈ 𝐸 such that 𝑥𝛼 ↑ 𝑥, i. e., 𝑥 = sup 𝑥𝛼 . For the proof of the next theorem we need the facts on 𝐴𝑀- and 𝐴𝐿-spaces provided in the Propositions 2.10 and 2.11. Theorem 4.10. The following statements hold: (1) let 𝐸 be an 𝐴𝐿-space and 𝐹 a Dedekind complete Banach lattice with an order unit. Then L 𝑟 (𝐸, 𝐹) has a rank one operator as an order unit and Φ𝑖 (L 𝑟 (𝐸, 𝐹)) = L 𝑟 (𝐸, 𝐹) for 𝑖 = 1, 2, 3;

6 since the cone 𝐸+ is generating. 7 see [59, Chapt. X.4] and [95, Def. 2.4.18].

76 | 4 Finite elements in vector lattices of linear operators (2) let 𝐸 and 𝐹 be Banach lattices with 𝐹 Dedekind complete. If Φ1 (L 𝑟 (𝐸, 𝐹)) = L 𝑟 (𝐸, 𝐹), then 𝐸 is lattice isomorphic to an 𝐴𝐿-space and 𝐹 is lattice isomorphic to an 𝐴𝑀-space; (3) let 𝐸 be a Dedekind complete Banach lattice. Then Φ1 (L 𝑟 (𝐸)) = L 𝑟 (𝐸) if and only if dim 𝐸 < ∞. Proof. (1) It is sufficient⁸ to prove only Φ1 (L 𝑟 (𝐸, 𝐹)) = L 𝑟 (𝐸, 𝐹). If 𝐸 is an 𝐴𝐿-space and 𝐹 is a Dedekind complete Banach lattice with an order unit 𝑒, then L 𝑟 (𝐸, 𝐹) is a vector space and has the order unit 𝑢󸀠 ⊗ 𝑒, where 𝑢󸀠 is an order unit in 𝐸󸀠 . Indeed, after passing to the equivalent order unit norm with respect to 𝑢󸀠 in 𝐸󸀠 , the closed unit ball of 𝐸󸀠 can be assumed to coincide with [−𝑢󸀠 , 𝑢󸀠 ] (see Proposition 2.10) and so, for 𝑇 ∈ L 𝑟 (𝐸, 𝐹) and 𝑥 ∈ 𝐸+ one has ‖𝑥‖ = 𝑢󸀠 (𝑥). Then |𝑇𝑥| ≤ ‖𝑇𝑥‖𝑒 ≤ ‖𝑇‖‖𝑥‖𝑒 = ‖𝑇‖𝑢󸀠 (𝑥)𝑒 = ‖𝑇‖(𝑢󸀠 ⊗ 𝑒)(𝑥) , which means |𝑇| ≤ ‖𝑇‖(𝑢󸀠 ⊗ 𝑒). Since the operator 𝑢󸀠 ⊗ 𝑒 is positive it obviously belongs to L 𝑟 (𝐸, 𝐹). So, the statement (1) is a consequence of Proposition 3.44. (2) Assume that Φ1 (L 𝑟 (𝐸, 𝐹)) = L 𝑟 (𝐸, 𝐹). Since L 𝑟 (𝐸, 𝐹) with the regular norm is a Banach lattice, by Theorem 3.19 it is lattice isomorphic to an 𝐴𝑀-space. It follows from the Theorems 2.2 and 3.2 of [136] and their proofs that 𝐸 is then lattice isomorphic to an 𝐴𝐿-space and 𝐹 is lattice isomorphic to an 𝐴𝑀-space, i. e., statement (2) holds. (3) According to Proposition 2.11 the statement (3) is a consequence of (2). Remark 4.11. 𝐹 may fail to have an order unit even if Φ1 (L 𝑟 (𝐸, 𝐹)) = L 𝑟 (𝐸, 𝐹), 𝐸 is an 𝐴𝐿-space and 𝐹 is an 𝐴𝑀-space. For example, let 𝐸 = ℝ and 𝐹 be the vector lattice as in Example 3.20 (see page 32). Then L 𝑟 (𝐸, 𝐹) = 𝐹, where 𝐸 is an 𝐴𝐿-space, 𝐹 a Dedekind complete 𝐴𝑀-space and Φ1 (L 𝑟 (𝐸, 𝐹)) = L 𝑟 (𝐸, 𝐹), but 𝐹 fails to have an order unit. Wickstead has characterized in [138] those Banach lattices 𝐸 for which L(𝐸, 𝐹) is a vector lattice for any (“universal” range) Banach lattice 𝐹. It is exactly the class of Banach lattices that are isomorphic to atomic 𝐴𝐿-spaces. It is then clear that L(𝐸, 𝐹) = L 𝑟 (𝐸, 𝐹) also holds (see [138, Theorem 2.4]). For such Banach lattices 𝐸 our theorem is applied to Φ1 (L(𝐸, 𝐹)) if the Banach lattice 𝐹 is Dedekind complete and arrives then at Φ1 (L(𝐸, 𝐹)) = Φ2 (L(𝐸, 𝐹)) = L(𝐸, 𝐹). There are provided other equivalent conditions for the relation 𝑟

L(𝐸, 𝐹) = L (𝐸, 𝐹)

⇐⇒

L(𝐸, 𝐹) is a vector lattice

(4.7)

to hold: (i) if 𝐹 is isomorphic to an 𝐴𝑀-space, the relation (4.7) is equivalent to the condition that 𝐸 is an atomic Banach lattice with order continuous norm (see [138, Theorem 2.9]).

8 see remark at page 26.

4.3 Finite rank operators in the vector lattice of regular operators

| 77

(ii) If 𝐸 is isomorphic to an 𝐴𝐿-space, the relation (4.7) is equivalent to the condition that 𝐹 has a Levi norm (see [138, Theorem 2.8]).

4.3 Finite rank operators in the vector lattice of regular operators Let 𝐸, 𝐹 be arbitrary Banach lattices. We denote the set of all finite rank operators 𝑇 : 𝐸 → 𝐹 by F(𝐸, 𝐹), i. e., 𝑚

󸀠

󸀠

󸀠

F(𝐸, 𝐹) = {𝑇 = ∑ 𝜓𝑖 ⊗ 𝜑𝑖 with 𝜓𝑖 ∈ 𝐸 , 𝜑𝑖 ∈ 𝐹 for some 𝑚, and 𝑖 = 1, . . . , 𝑚}, 𝑖=1

and now study finite rank operators, in particular, rank one operators in L 𝑟 (𝐸, 𝐹). It is clear that each finite rank operator is continuous, even L 𝑟 (𝐸, 𝐹) ⊂ L(𝐸, 𝐹). If 𝑇 = 𝜓󸀠 ⊗ 𝜑 ∈ L 𝑟 (𝐸, 𝐹) is an arbitrary rank one operator, where 𝜓󸀠 ∈ 𝐸󸀠 , 𝜑 ∈ 𝐹, then its modulus exists and is the (likewise rank one) operator |𝑇| = |𝜓󸀠 | ⊗ |𝜑| (see [9, Theorem 5.7]). Notice that each finite rank operator possesses a (compact) modulus, which is not necessarily of finite rank (see [9, Theorem 16.8], [111, § III.3]). For a finite rank operator 󸀠 𝑇 = ∑𝑚 𝑖=1 𝜓𝑖 ⊗ 𝜑𝑖 , the operator 𝑚

𝑇3 = ∑ |𝜓𝑖󸀠 | ⊗ |𝜑𝑖 |, 𝑖=1

which is natural to be considered, satisfies the inequality |𝑇| ≤ 𝑇3 ; however it might not coincide with |𝑇| if 𝑚 ≥ 2, as shown in the next example. Example 4.12. Let 𝐸 = 𝐹 = ℝ2 with 𝐸+ = ℝ2+ , and 𝑒1 = (1, 0), 𝑒2 = (0, 1). For 𝜑1 = 𝑒1 , 𝜑2 = 𝑒1 + 𝑒2 = (1, 1), 𝜓1󸀠 = 𝑒2 − 𝑒1 = (−1, 1), 𝜓2󸀠 = 𝑒1 consider the operator 𝑇 = 𝜓1󸀠 ⊗ 𝜑1 + 𝜓2󸀠 ⊗ 𝜑2 . Then the following facts are obvious: (i) each one of the sets {𝜑1 , 𝜑2 }, and {𝜓1󸀠 , 𝜓2󸀠 }, consists of linearly independent vectors; (ii) 𝑇 = 𝑒2 ⊗ 𝑒1 + 𝑒1 ⊗ 𝑒2 ≥ 0, so that 𝑇 = |𝑇|; (iii) 𝑇3 = |𝜓1󸀠 | ⊗ |𝜑1 | + |𝜓2󸀠 | ⊗ |𝜑2 | = 𝑒2 ⊗ 𝑒1 + 2(𝑒1 ⊗ 𝑒1 ) + 𝑒1 ⊗ 𝑒2 > |𝑇|. 󸀠 In the cases when the sets {𝜑1 , . . . , 𝜑𝑚 }, or {𝜓1󸀠 , . . . , 𝜓𝑚 }, (or both), consist of pairwise disjoint elements, one has the following result (see [111, Lemma 3.4]).

Proposition 4.13. Let 𝐸 and 𝐹 be Banach lattices and 𝑇 : 𝐸 → 𝐹 the operator 𝑇 = 󸀠 ∑𝑚 𝑖=1 𝜓𝑖 ⊗ 𝜑𝑖 . Then the following statements hold: (1) if the elements 𝜑1 , . . . , 𝜑𝑚 ∈ 𝐹 are pairwise disjoint then |𝑇| = 𝑇3 ; 󸀠 (2) if the elements 𝜓1󸀠 , . . . , 𝜓𝑚 ∈ 𝐸󸀠 are pairwise disjoint then |𝑇| = 𝑇3 . The next result provides some necessary and sufficient conditions for a rank one operator to be a finite element in L 𝑟 (𝐸, 𝐹). Notice that in contrast to the situation we

78 | 4 Finite elements in vector lattices of linear operators face later in Section 4.6, further on in this section we assume the vector lattice 𝐹 to be Dedekind complete. This not only guarantees the vector lattice structure for L 𝑟 (𝐸, 𝐹), but also allows proof of the sufficiency part in Theorem 4.14. Theorem 4.14. Let 𝐸 and 𝐹 be Banach lattices such that 𝐹 is Dedekind complete, and let the rank one operator 𝑇 = 𝜓󸀠 ⊗ 𝜑 belong to L 𝑟 (𝐸, 𝐹). Then 𝑇 is finite in L 𝑟 (𝐸, 𝐹), i. e. , 𝑇 ∈ Φ1 (L 𝑟 (𝐸, 𝐹)), if and only if 𝜓󸀠 ∈ Φ1 (𝐸󸀠 ), and 𝜑 ∈ Φ1 (𝐹). Proof. Necessity. Clearly, only 𝑇 ≠ 0 is of interest. Due to |𝑇| = |𝜓󸀠 | ⊗ |𝜑|, we may assume that 𝜓󸀠 > 0, and 𝜑 > 0 . If 𝑇 is a finite element in L 𝑟 (𝐸, 𝐹), and 𝑈 ∈ L+ (𝐸, 𝐹) is one of its majorants, then for each 𝑆 ∈ L 𝑟 (𝐸, 𝐹) there is a positive number 𝑐𝑆 , such that |𝑆| ∧ 𝑛 𝑇 ≤ 𝑐𝑆 𝑈 for all 𝑛 ∈ ℕ. (4.8) For any ℎ ∈ 𝐹 consider the operator⁹ 𝑆 = 𝜓󸀠 ⊗ ℎ ∈ L 𝑟 (𝐸, 𝐹). Then from the last inequality and Theorem 1.16 of [9] it follows that 𝑘

𝑘

𝑐𝑆 𝑈𝑥 ≥ (|𝑆| ∧ 𝑛 𝑇)(𝑥) = inf { ∑(|𝑆|𝑥𝑖 ) ∧ (𝑛 𝑇𝑥𝑖 ) : 𝑥𝑖 ∈ 𝐸+ , ∑ 𝑥𝑖 = 𝑥, 𝑛 ∈ ℕ} 𝑘

𝑖=1

𝑖=1

𝑘

𝑘

= inf { ∑(𝜓󸀠 (𝑥𝑖 )|ℎ|) ∧ (𝜓󸀠 (𝑥𝑖 )𝑛𝜑) : 𝑥𝑖 ∈ 𝐸+ , ∑ 𝑥𝑖 = 𝑥, 𝑛 ∈ ℕ} 𝑘

𝑖=1

𝑖=1

𝑘

𝑘

𝑖=1

𝑖=1

= inf { ∑ 𝜓󸀠 (𝑥𝑖 )(|ℎ| ∧ (𝑛𝜑)) : 𝑥𝑖 ∈ 𝐸+ , ∑ 𝑥𝑖 = 𝑥, 𝑛 ∈ ℕ} 𝑘

󸀠

= 𝜓 (𝑥)(|ℎ| ∧ (𝑛𝜑)) for all 𝑥 ∈ 𝐸+ and 𝑛 ∈ ℕ. Choose 𝑥0 ∈ 𝐸+ such that 𝜓󸀠 (𝑥0 ) = 1. Then for all 𝑛 ∈ ℕ,

|ℎ| ∧ (𝑛𝜑) ≤ 𝑐𝑆 𝑈𝑥0

which implies that 𝜑 is a finite element in 𝐹 with 𝑈𝑥0 as one of its majorants. On the other hand, for any ℎ󸀠 ∈ 𝐸󸀠 consider the operator 𝑆 = ℎ󸀠 ⊗ 𝜑. Then again due to the Inequality (4.8) for arbitrary 𝑥 and 𝑥𝑖 ∈ 𝐸+ with ∑𝑘𝑖=1 𝑥𝑖 = 𝑥 , this time one has (|ℎ󸀠 | ∧ 𝑛𝜓󸀠 )(𝑥𝑖 ) 𝜑 ≤ (|ℎ󸀠 |(𝑥𝑖 ) 𝜑) ∧ (𝑛𝜓󸀠 (𝑥𝑖) 𝜑) = (|𝑆|𝑥𝑖 ) ∧ (𝑛𝑇𝑥𝑖 ) for each 1 ≤ 𝑖 ≤ 𝑘. Thus 𝑘

(|ℎ󸀠 | ∧ 𝑛𝜓󸀠 )(𝑥) 𝜑 ≤ ∑(|𝑆|𝑥𝑖 ) ∧ (𝑛 𝑇𝑥𝑖 ) . 𝑖=1

9 The case 𝑆 = 0 is trivial.

4.3 Finite rank operators in the vector lattice of regular operators |

79

Since this estimate holds for arbitrary decompositions of the element 𝑥, it is also true for the infimum of these sums. Together with the Inequality 4.8, one arrives at (|ℎ󸀠 | ∧ 𝑛𝜓󸀠 )(𝑥) 𝜑 ≤ (|𝑆| ∧ (𝑛 𝑇)) (𝑥) ≤ 𝑐𝑆 𝑈𝑥 for all 𝑥 ∈ 𝐸+ and 𝑛 ∈ ℕ. Choose 𝑢󸀠 ∈ 𝐹+󸀠 such that 𝑢󸀠 (𝜑) = 1, then (|ℎ󸀠 | ∧ 𝑛𝜓󸀠 )(𝑥) ≤ 𝑢󸀠 (𝑐𝑆 𝑈𝑥) = 𝑐𝑆 (𝑈󸀠 𝑢󸀠 )(𝑥) 𝑥 ∈ 𝐸+ , 𝑛 ∈ ℕ. Therefore, |ℎ󸀠 | ∧ 𝑛𝜓󸀠 ≤ 𝑐𝑆 (𝑈󸀠 𝑢󸀠 ) = 𝑐𝑆 𝑣󸀠 󸀠

󸀠 󸀠

𝐸󸀠+ . This

for all 𝑛 ∈ ℕ ,

󸀠

shows that 𝜓 is finite in 𝐸󸀠 with 𝑣󸀠 as one of its majorants. where 𝑣 = 𝑈 𝑢 ∈ Sufficiency. For the finite elements 𝜓󸀠 ∈ 𝐸󸀠 and 𝜑 ∈ 𝐹, we may assume 𝜓󸀠 > 0 and 𝜑 > 0, so 𝑇 = 𝜓󸀠 ⊗ 𝜑 > 0. Then by Theorem 3.11 the bands {𝜑}⊥⊥ and {𝜓󸀠 }⊥⊥ have order units, say 𝑒 and 𝑢󸀠 , respectively. Due to the Dedekind completeness of 𝐹, it follows from Theorem 4.4 that, without loss of generality, we may assume 𝐹 = {𝜑}⊥⊥ . We also suppose that 𝐹 is equipped with the order unit norm. For any 𝑥󸀠 ∈ {𝜓󸀠 }⊥⊥ with ‖𝑥󸀠 ‖ ≤ 1 there is a constant 𝜆 > 0 such that |𝑥󸀠 | ≤ 𝜆𝑢󸀠 (cf. Theorem 3.15). Now we claim that the operator 𝑈 = 𝑢󸀠 ⊗ 𝑒 is a majorant of 𝑇, which certainly implies that 𝑇 is a finite element in L 𝑟 (𝐸, 𝐹). Indeed, for any 𝑆 ∈ L+ (𝐸, 𝐹), it is clear that (𝑆 ∧ 𝑛 𝑇)󸀠 : 𝐹󸀠 → {𝜓󸀠 }𝑑𝑑 ⊂ 𝐸󸀠 , as (𝑆 ∧ 𝑛 𝑇)󸀠 ≤ 𝑛 𝑇󸀠 , and 𝑇󸀠 (𝐹󸀠 ) ⊂ {𝜓󸀠 }𝑑𝑑 . It follows that for each 𝑦󸀠 ∈ 𝐹󸀠 |(𝑆 ∧ 𝑛 𝑇)󸀠 𝑦󸀠 | ≤ (𝑆 ∧ 𝑛 𝑇)󸀠 |𝑦󸀠 | ≤ 𝜆‖(𝑆 ∧ 𝑛 𝑇)󸀠 |𝑦󸀠 |‖ 𝑢󸀠 ≤ 𝜆‖𝑆‖‖𝑦󸀠 ‖ 𝑢󸀠 as (𝑆 ∧ 𝑛 𝑇)󸀠 𝑦󸀠 ∈ {𝜓󸀠 }𝑑𝑑. Since ‖𝑦󸀠 ‖ = |𝑦󸀠 |(𝑒), we have (𝑆 ∧ 𝑛 𝑇)󸀠 𝑦󸀠 ≤ 𝜆‖𝑆‖𝑦󸀠 (𝑒) 𝑢󸀠

for 𝑦󸀠 ∈ 𝐹+󸀠 ,

which, together with 𝑢󸀠 (𝑥)𝑦󸀠(𝑒) = 𝑦󸀠 (𝑈𝑥), implies that 𝑦󸀠 ((𝑆 ∧ 𝑛 𝑇)𝑥) = ((𝑆 ∧ 𝑛 𝑇)󸀠 𝑦󸀠 )(𝑥) ≤ 𝜆‖𝑆‖𝑦󸀠 (𝑒) 𝑢󸀠 (𝑥) = 𝑦󸀠 (𝜆‖𝑆‖𝑈𝑥) for all 𝑥 ∈ 𝐸+ and 𝑦󸀠 ∈ 𝐹+󸀠 . Therefore (𝑆 ∧ 𝑛 𝑇)𝑥 ≤ 𝜆‖𝑆‖𝑢󸀠 (𝑥) = 𝜆‖𝑆‖𝑈𝑥 for all 𝑥 ∈ 𝐸+ , i. e. , 𝑆 ∧ 𝑛 𝑇 ≤ 𝜆‖𝑆‖𝑈 for all 𝑛 ∈ ℕ as claimed. Corollary 4.15. Let 𝐸 and 𝐹 be Banach lattices such that 𝐹 is Dedekind complete, and 𝑚 let 𝑇 = ∑𝑖=1 𝜓𝑖󸀠 ⊗ 𝜑𝑖 be a finite rank operator which belongs to L 𝑟 (𝐸, 𝐹). If 𝜓𝑖󸀠 ∈ Φ1 (𝐸󸀠 ) and 𝜑𝑖 ∈ Φ1 (𝐹) for 𝑖 = 1, . . . , 𝑚, then 𝑇 is finite in L 𝑟 (𝐸, 𝐹) with a rank one operator as one of its majorants. Proof. The assertion of the corollary follows immediately from the inequality |𝑇| ≤ 𝜓󸀠 ⊗ 󸀠 𝜑, where 𝜓󸀠 = |𝜓1󸀠 |+|𝜓2󸀠 |+⋅ ⋅ ⋅+|𝜓𝑚 | ∈ Φ1 (𝐸󸀠 ), and 𝜑 = |𝜑1 |+|𝜑2 |+⋅ ⋅ ⋅+|𝜑𝑚 | ∈ Φ1 (𝐹), since Φ1 (L 𝑟 (𝐸, 𝐹)) is an ideal in L 𝑟 (𝐸, 𝐹) and, due to the previous theorem, the operator 𝜓󸀠 ⊗ 𝜑 belongs to Φ1 (L 𝑟 (𝐸, 𝐹)).

80 | 4 Finite elements in vector lattices of linear operators 𝑚

For finite rank operators 𝑇 = ∑𝑖=1 𝜓𝑖󸀠 ⊗ 𝜑𝑖 ∈ L 𝑟 (𝐸, 𝐹) such that either the collection {𝜑𝑖 ∈ 𝐹 : 1 ≤ 𝑖 ≤ 𝑚}, or {𝜓𝑖󸀠 ∈ 𝐸󸀠 : 1 ≤ 𝑖 ≤ 𝑚} (or both), consist of pairwise disjoint elements, the modulus of 𝑇 is known from Proposition 4.13. This enables us to prove the converse statement to the last corollary in these two particular cases. Corollary 4.16. Let 𝐸 and 𝐹 be Banach lattices such that 𝐹 is Dedekind complete, and 𝑚 let 𝑇 = ∑𝑖=1 𝜓𝑖󸀠 ⊗ 𝜑𝑖 ∈ L 𝑟 (𝐸, 𝐹), where 𝜑1 , 𝜑2 , . . . , 𝜑𝑚 are pairwise disjoint elements of 𝐹. If 𝑇 is a finite element in L 𝑟 (𝐸, 𝐹), then 𝜓𝑖󸀠 ∈ Φ1 (𝐸󸀠 ), and 𝜑𝑖 ∈ Φ1 (𝐹) for 1 ≤ 𝑖 ≤ 𝑚. Proof. One has

𝑚

|𝑇| = ∑ |𝜓𝑖󸀠 | ⊗ |𝜑𝑖 | and

|𝑇| ≥ 𝜓𝑖󸀠 ⊗ 𝜑𝑖

𝑖=1

for each 1 ≤ 𝑖 ≤ 𝑚. Since Φ1 (L 𝑟 (𝐸, 𝐹)) is an ideal in L 𝑟 (𝐸, 𝐹), for 1 ≤ 𝑖 ≤ 𝑚 the operators 𝜓𝑖󸀠 ⊗𝜑𝑖 also belong to Φ1 (L 𝑟 (𝐸, 𝐹)) as |𝑇| together with 𝑇 is finite in L 𝑟 (𝐸, 𝐹). Now the assertion follows from Theorem 4.14. In a similar way, again guided by Proposition 4.13, we get Corollary 4.17. Let 𝐸 and 𝐹 be Banach lattices such that 𝐹 is Dedekind complete, and 󸀠 𝑟 󸀠 󸀠 󸀠 let 𝑇 = ∑𝑚 𝑖=1 𝜓𝑖 ⊗ 𝜑𝑖 ∈ L (𝐸, 𝐹), where 𝜓1 , 𝜓2 , . . . , 𝜓𝑚 are pairwise disjoint elements of 󸀠 𝑟 󸀠 𝐸 . If 𝑇 is a finite element in L (𝐸, 𝐹), then 𝜓𝑖 ∈ Φ1 (𝐸󸀠 ) and 𝜑𝑖 ∈ Φ1 (𝐹) for 1 ≤ 𝑖 ≤ 𝑚. If the constituent parts of a finite rank operator are positive, then we have another converse result to Corollary 4.15 Corollary 4.18. Let 𝐸 and 𝐹 be Banach lattices such that 𝐹 is Dedekind complete, and 󸀠 𝑟 󸀠 let 𝑇 = ∑𝑚 𝑖=1 𝜓𝑖 ⊗ 𝜑𝑖 ∈ L (𝐸, 𝐹) be such that 𝜓𝑖 ≥ 0, 𝜑𝑖 ≥ 0, 𝑖 = 1, . . . , 𝑚. If 𝑇 is a finite element in L 𝑟 (𝐸, 𝐹), then 𝜓𝑖󸀠 ∈ Φ1 (𝐸󸀠 ), and 𝜑𝑖 ∈ Φ1 (𝐹) for 1 ≤ 𝑖 ≤ 𝑚. Note that the proof is the same as in Corollary 4.16. Since the representation of a finite rank operator is not unique, the following example shows that the converse of Corollary 4.15, in general, is false, even if the sys󸀠 tems, either {𝜓1󸀠 , ⋅ ⋅ ⋅ , 𝜓𝑚 } or {𝜑1 , ⋅ ⋅ ⋅ , 𝜑𝑚 }, are linear independent. Example 4.19. Let 𝐸 = 𝐹 = ℓ2 , 𝜑 = (1, 12 , 13 , ⋅ ⋅ ⋅ ), 𝜑𝑛 = (1, ⋅ ⋅ ⋅ , 1𝑛 , 0, ⋅ ⋅ ⋅ ), and 𝛼𝑛 = 𝜑 − 𝜑𝑛 . Then 𝜑, 𝜑𝑛 , 𝛼𝑛 ∈ 𝐸 = 𝐸󸀠 . Define for some fixed 𝑛 ≥ 2 the operator 𝑇 = 𝜑𝑛 ⊗ 𝜑 + (−𝜑𝑛 ) ⊗ 𝛼𝑛 = 𝜑 ⊗ 𝜑𝑛 + 𝛼𝑛 ⊗ (−𝜑𝑛 ) = 𝜑𝑛 ⊗ 𝜑𝑛 . Then (i) 𝜑𝑛 is finite in 𝐸 and 𝐸󸀠 (see the Remark after Theorem 3.18), but 𝜑 and 𝛼𝑛 are not finite; (ii) 𝑇 is finite in L 𝑟 (𝐸) as 𝑇 = 𝜑𝑛 ⊗ 𝜑𝑛 (by Theorem 4.14); (iii) from the above representations of 𝑇, it is clear that neither 𝜑 nor 𝛼𝑛 are finite elements in 𝐸 or 𝐸󸀠 , although 𝜑 and 𝛼𝑛 are linearly independent. This shows that the converse of Corollary 4.15 is false.

4.4 Some vector lattices and Banach lattices of operators |

81

The following question is still open: Let 𝐸 and 𝐹 be Banach lattices such that 𝐹 is Dedekind complete, and let 𝑇 : 𝐸 → 𝐹 be a finite rank operator which is known to be a finite element in L 𝑟 (𝐸, 𝐹). Do 𝜓𝑖󸀠 ∈ Φ1 (𝐸󸀠 ) and 𝜑𝑖 ∈ Φ1 (𝐹) (1 ≤ 𝑖 ≤ 𝑚) exist for some 󸀠 𝑚 such that 𝑇 = ∑𝑚 𝑖=1 𝜓𝑖 ⊗ 𝜑𝑖 ? Corollary 4.20. Let 𝐸 and 𝐹 be Banach lattices such that 𝐹 is Dedekind complete. If 𝑇 ∈ L 𝑟 (𝐸, 𝐹) is finite, then 𝑇 maps atoms of 𝐸 into finite elements in 𝐹. Proof. If 𝑎 ∈ 𝐸+ is an atom, then it is easy to verify that {𝑎}⊥⊥ = {𝜆𝑎: 𝜆 ∈ ℝ} is a projection band ([84, Theorem 26.4]), moreover 𝑃𝑎 𝑥 = 𝜆 𝑎 (𝑥)𝑎 for each 𝑥 ∈ 𝐸+ , where 𝑃𝑎 : 𝐸 → {𝑎}⊥⊥ is the band projection, and 𝜆 𝑎 (𝑥) = sup{𝑟 ∈ ℝ+ : 𝑟𝑎 ≤ 𝑥}. Define the operator 𝑆 : {𝑎}⊥⊥ → 𝐹 by 𝑆(𝜆𝑎) = 𝜆|𝑇𝑎|. It is easy to verify that 𝑆𝑃𝑎 ∈ L 𝑟 (𝐸, 𝐹), and since 𝑥 ∈ 𝐸+ and 𝑃𝑎 𝑥 = 𝜆 𝑎 (𝑥)𝑥 ≤ 𝑥 imply (𝑆𝑃𝑎 )(𝑥) = 𝑆(𝑃𝑎 𝑥) = 𝑆(𝜆 𝑎 (𝑥)𝑎) = 𝜆 𝑎 (𝑥)|𝑇𝑎| = |𝑇(𝜆 𝑎 (𝑥)𝑎)| ≤ |𝑇|(𝜆 𝑎 (𝑥)𝑎) ≤ |𝑇|(𝑥), one has 𝑆𝑃𝑎 ≤ |𝑇|. Thus 𝑆𝑃𝑎 is finite in L 𝑟 (𝐸, 𝐹), as Φ1 (L 𝑟 (𝐸, 𝐹)) is an ideal of L 𝑟 (𝐸, 𝐹). Obviously 𝑆𝑃𝑎 is a rank one operator, therefore there is some 𝜓󸀠 ∈ 𝐸󸀠 such that 𝑆𝑃𝑎 = 𝜓󸀠 ⊗ |𝑇𝑎|. It follows from the previous theorem that 𝑇𝑎 is a finite element in 𝐹. Now we are able to provide the example mentioned earlier in Remark 4.3. Example 4.21. Let 𝐸 = ℓ𝑝 , and 𝐹 = 𝐿 𝑝 [0, 1] with 1 ≤ 𝑝 < ∞. The vector lattices 𝐸 and 𝐹 are Dedekind complete (see [120, § IV.1]). Take a disjoint sequence (𝑥𝑛) ⊂ 𝐹+ with ‖𝑥𝑛 ‖ = 1, and define the operator 𝑇 : 𝐸 → 𝐹 by 𝑇(𝜆 𝑛) = ∑∞ 𝑛=1 𝜆 𝑛 𝑥𝑛 . Then 𝑇 is an isometric lattice isomorphism from 𝐸 onto 𝑇(𝐸). Observe that this example stands in contrast to Corollary 4.20. By that corollary the operator 𝑇 is not finite, since 𝑇 maps the atoms (finite elements) 𝑒𝑛 into 𝑥𝑛 , which are not finite elements in 𝐹 as Φ1 (𝐹) = {0} (see Remark after Theorem 3.18). Here 𝑒𝑛 denotes the sequence (in ℓ𝑝 ) with its 𝑛-th entry equal to 1, and all others are 0.

4.4 Some vector lattices and Banach lattices of operators For our subsequent investigations of operators to be finite elements we need ambient vector lattices of operators. Some of them will be provided in the next two subsections before the main problem can be addressed in Sections 4.5–4.6. The vector space L 𝑟 (𝐸, 𝐹) of all regular operators between the vector lattices 𝐸, 𝐹, and some of its subspaces might also be vector lattices in a more general situation when 𝐹 is not necessarily Dedekind complete; see Wickstead’s results, mentioned at the end of Section 4.2. In this section, following the paper [54], we deal with ordered vector spaces V(𝐸, 𝐹) of linear operators and ask under which conditions are they vector lattices, latticesubspaces of the ordered vector space L 𝑟 (𝐸, 𝐹) or, in the case that L 𝑟 (𝐸, 𝐹) is a vector

82 | 4 Finite elements in vector lattices of linear operators lattice, sublattices or even Banach lattices when equipped with the regular norm. For many classes of operators acting between appropriate Banach lattices the answer is affirmative, e. g., for compact, weakly compact, regular 𝐴𝑀-compact, regular Dunford– Pettis operators, and others. Then it is possible to study the finite elements in such vector lattices V(𝐸, 𝐹), where 𝐹 is not necessarily Dedekind complete. Definition 4.22. A Banach lattice 𝐸 is said to be a 𝐾𝐵-space if every increasing normbounded sequence of 𝐸+ converges with respect to the norm, i. e., 0 ≤ 𝑥𝑛 ↑, and sup ‖𝑥𝑛 ‖ < ∞ imply that (𝑥𝑛 ) is norm-convergent. Regarding the properties of a 𝐾𝐵-space 𝐸, we mention only some of them. The norm in 𝐸 is order continuous¹⁰, and therefore 𝐸 is Dedekind complete. Any 𝐾𝐵-space 𝐸 is weakly sequentially complete, i. e., every weak Cauchy sequence in 𝐸 converges weakly to some element in 𝐸. Any 𝐾𝐵-space 𝐸 is a band in 𝐸󸀠󸀠 . A Dedekind complete Banach lattice 𝐸 is a 𝐾𝐵-space if and only if it satisfies the conditions (A) and (B), where both conditions¹¹ are supposed to hold only for sequences (see [59, § X.4.4]). Every 𝐴𝐿-space is a 𝐾𝐵-space (see [120, Theorem VII.7.1], [9, Sect. 14]). The norm dual 𝐸󸀠 of a Banach lattice 𝐸 is a 𝐾𝐵-space if and only if 𝐸󸀠 has order continuous norm (see [9, 59, 95]). Recall that a Banach lattice 𝐸 is said to have the property (W1), if for each relatively weakly compact subset 𝐴 ⊂ 𝐸, the set |𝐴| = {|𝑥| : 𝑥 ∈ 𝐴} is also relatively weakly compact. Any 𝐾𝐵-space has the property (W1) (see [41, 52]). Here we repeat some facts which have been mentioned already earlier. If 𝐸 is a Banach lattice with an order unit, then the corresponding order unit norm is an equivalent norm on 𝐸, and 𝐸 with the order unit norm is an 𝐴𝑀-space with order unit (see p. 14). If 𝐸 and 𝐹 are normed vector lattices, then the space L(𝐸, 𝐹) of all linear continuous operators from 𝐸 into 𝐹 is equipped with the order defined by the positive operators of L (𝐸, 𝐹), and the standard operator norm (4.1). Every positive operator from a Banach lattice 𝐸 into a normed vector lattice 𝐹 is continuous (see Theorem 2.12), where its norm can be calculated by using in (4.1) only the positive elements of the unit ball. If 𝐹 is a Banach lattice, then the space L 𝑟 (𝐸, 𝐹) with its regular norm (4.2) is a Banach space. If the Banach lattice 𝐹 is Dedekind complete, then ‖𝑇‖𝑟 is a Riesz norm which can be calculated by ‖𝑇‖𝑟 = ‖|𝑇|‖

for all

𝑇 ∈ L 𝑟 (𝐸, 𝐹),

and (L 𝑟 (𝐸, 𝐹), ‖⋅‖𝑟 ) is a (Dedekind complete) Banach lattice (see [2, p. 22], [9, Theorem 15.2] and [95, Proposition 1.3.6]). Of course, the last formula makes sense for any

10 I. e. , satisfies the condition (A): 0 ≤ 𝑥𝛼 ↓ 0 implies ‖𝑥𝛼 ‖ 󳨀→ 0; see p. 13. 11 Condition (B): 0 ≤ 𝑥𝛼 ↑ and sup ‖𝑥𝛼 ‖ < ∞ imply 𝑥𝛼 ↑ 𝑥 for some 𝑥 ∈ 𝐸, (i. e., ‖⋅‖ is a Levi norm); see p. 75.

4.4 Some vector lattices and Banach lattices of operators

| 83

operator possessing a modulus, and later on it will be used to introduce a Riesz norm in vector lattices of operators; see e. g., Theorem 4.25.

4.4.1 Vector lattices of operators Further on we are interested in vector spaces G(𝐸, 𝐹) of (linear) operators between vector lattices 𝐸 and 𝐹, where the order in G(𝐸, 𝐹) is introduced by the wedge G(𝐸, 𝐹) ∩ L+(𝐸, 𝐹) of its positive operators. It is well known that for some ordered vector spaces G(𝐸, 𝐹), an operator 𝑇 ∈ G(𝐸, 𝐹) may not have its modulus (in G(𝐸, 𝐹)) or has the modulus |𝑇|, when 𝑇 is considered as an element in a larger space, for example in the space L (𝐸, 𝐹) of all linear operators between 𝐸 and 𝐹. But then |𝑇| ∉ G(𝐸, 𝐹) may happen. If for 𝑇 ∈ G(𝐸, 𝐹) the modulus exists in G(𝐸, 𝐹), then |𝑇𝑥| ≤ |𝑇|(|𝑥|)

for any

(4.9)

𝑥 ∈ 𝐸.

It is clear that if G(𝐸, 𝐹) is a vector lattice of linear operators, then each operator 𝑇 ∈ G(𝐸, 𝐹) possesses its modulus (in G(𝐸, 𝐹)), which in turn shows that 𝑇 is regular, and therefore G(𝐸, 𝐹) ⊆ L 𝑟 (𝐸, 𝐹). Only a few situations are known for L 𝑟 (𝐸, 𝐹) to be a vector lattice. As already mentioned, by the Riesz-Kantorovich Theorem (see 2.4) one has L 𝑏 (𝐸, 𝐹) = L 𝑟 (𝐸, 𝐹), and L 𝑟 (𝐸, 𝐹) is a Dedekind complete vector lattice if 𝐹 is Dedekind complete. Then the modulus of an operator 𝑇 can be calculated by the Riesz-Kantorovich formula (see p. 9): |𝑇|(𝑥) = |𝑇| (𝑥) = sup{𝑇𝑦 : − 𝑥 ≤ 𝑦 ≤ 𝑥} = sup{|𝑇𝑦| : |𝑦| ≤ 𝑥}

for each 𝑥 ∈ 𝐸+ . (4.10)

For a Banach lattice 𝐸, the sequential weak*-continuity of the lattice operations in 𝐸󸀠 is a necessary and sufficient condition for the space L 𝑟 (𝐸, c) to be a vector lattice (see [105]), where c is the vector lattice of all real converging sequences which is not even 𝜎-Dedekind complete. Important in this connection is the following property of a Banach lattice. Definition 4.23. A Banach lattice 𝐸 has the property (∗) if, whenever a sequence (𝑓𝑛 ) ⊂ 󵄨 󵄨 𝐸󸀠+ , which converges in the topology 𝜎(𝐸󸀠 , 𝐸) to 𝑓 ∈ 𝐸󸀠 as 𝑛 → ∞, one has 󵄨󵄨󵄨𝑓𝑛 − 𝑓󵄨󵄨󵄨 → 0 󸀠 in 𝜎(𝐸 , 𝐸) as 𝑛 → ∞. A Banach lattice 𝐸 has the property (∗) if and only if L 𝑟 (𝐸, c) is a vector lattice. If 𝐸 and 𝐹 are Banach lattices and L 𝑟 (𝐸, 𝐹) is a vector lattice then either 𝐸 has property (∗) or 𝐹 is 𝜎-Dedekind complete (see [34, 105, 138]). In [34, Proposition 1.2.] it is proved that for any atomic Banach lattice 𝐸 with order continuous norm, and each Banach lattice 𝐹, the space L 𝑏 (𝐸, 𝐹) is a vector lattice, and L 𝑟 (𝐸, 𝐹) = L 𝑏 (𝐸, 𝐹). If 𝐸 is an 𝐴𝐿-space and 𝐹 a 𝐾𝐵-space then, according to [9, Theorem 15.3], one also has L(𝐸, 𝐹) = L 𝑏 (𝐸, 𝐹), and since 𝐹 is Dedekind complete, L(𝐸, 𝐹) is a vector lattice.

84 | 4 Finite elements in vector lattices of linear operators Based on the fact that in a Banach lattice with order unit the collections of normbounded and order bounded subsets coincide, it is easy to prove that L(𝐸, 𝐹) = L 𝑏 (𝐸, 𝐹), provided 𝐸, 𝐹 are Banach lattices, and 𝐹 has an order unit. The main properties of the finite elements in the vector lattice L 𝑟 (𝐸, 𝐹) were already described in our Theorems 4.2 and 4.4 for the case that 𝐸 is a Banach lattice and 𝐹 is a Dedekind complete Banach lattice. Now we provide a formula for the modulus of an operator in an arbitrary vector lattice of operators. Proposition 4.24. Let 𝐸, 𝐹 be vector lattices. If V(𝐸, 𝐹) is an arbitrary vector lattice of linear operators, then for 𝑇 ∈ V(𝐸, 𝐹) one has |𝑇| = inf{𝑆 ∈ V+ (𝐸, 𝐹) : |𝑇𝑥| ≤ 𝑆|𝑥| for all 𝑥 ∈ 𝐸}. Proof. Let 𝑆 ∈ V+ (𝐸, 𝐹) such that |𝑇𝑥| ≤ 𝑆|𝑥|, 𝑥 ∈ 𝐸, where due to (4.9) the modulus |𝑇| is such an operator. Then ±𝑇𝑥 ≤ 𝑆𝑥 for 𝑥 ∈ 𝐸+ , i. e., ±𝑇 ≤ 𝑆, and so 𝑆 is an upper bound for the set {−𝑇, 𝑇} in V(𝐸, 𝐹). This implies |𝑇| ≤ 𝑆, i. e., |𝑇| is the smallest element in the considered set. The most interesting class, besides the regular and continuous operators, is the ordered vector space K(𝐸, 𝐹) of all compact operators between the Banach lattices 𝐸 and 𝐹, i. e., operators which map any norm-bounded subset of 𝐸 onto a relatively norm compact subset of 𝐹. It is well known that it may not be a vector lattice, even if both 𝐸 and 𝐹 are Dedekind complete. In [4] and [5] it is shown that, in particular, regular compact operators exist which do not possess a modulus, and that a compactly dominated compact operator 𝑆 (even for Dedekind complete 𝐸 = 𝐹) exists, with modulus in L(𝐸) such that |𝑆| is not compact. So in general, K(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹) is a proper subspace of K(𝐸, 𝐹), which also may fail to be a vector lattice even if 𝐹 is Dedekind complete. The subspace K𝑟 (𝐸, 𝐹) of K(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹), generated by the positive compact operators, i. e., K𝑟 (𝐸, 𝐹) = K+ (𝐸, 𝐹) − K+ (𝐸, 𝐹), is exhaustively studied in [135], where necessary and sufficient conditions on 𝐸 and 𝐹 are found for K𝑟 (𝐸, 𝐹) to be a 𝜎-Dedekind complete, or even a Dedekind complete vector lattice (see Remark 4.29 (2) below).

4.4.2 Banach lattices of operators It has already been pointed out that in several situations, e. g., in Sections 3.2–3.4, and in the preceding sections of the present chapter, the study of finite elements in vector lattices makes essential use of the structure of a Banach lattice (see also [36– 38]). When dealing with a vector lattice V(𝐸, 𝐹) of operators between Banach lattices 𝐸 and 𝐹, one has V(𝐸, 𝐹) ⊆ L 𝑟 (𝐸, 𝐹) ⊆ L(𝐸, 𝐹), and so V(𝐸, 𝐹) can be equipped with the operator norm (4.1). However, in general the operator norm is not a Riesz norm, and therefore for our purposes it will be favourable to equip the vector lattice V(𝐸, 𝐹) with the regular norm (4.2).

4.4 Some vector lattices and Banach lattices of operators |

85

Theorem 4.25. Let 𝐸, 𝐹 be Banach lattices. If V(𝐸, 𝐹) is an arbitrary vector lattice of linear continuous operators. Then (1) ‖𝑇‖𝑟 = ‖ |𝑇| ‖ is a Riesz norm on V(𝐸, 𝐹), where ‖𝑇‖ ≤ ‖𝑇‖𝑟 ; (2) ‖𝑇‖𝑟 = inf{‖𝑆‖ : 𝑆 ∈ V+ (𝐸, 𝐹), |𝑇𝑥| ≤ 𝑆|𝑥| for all 𝑥 ∈ 𝐸} and (V(𝐸, 𝐹), ‖ ⋅ ‖𝑟 ) is a normed vector lattice. Proof. (1) We omit the standard proof that ‖𝑇‖𝑟 = ‖|𝑇|‖ defines a norm on V(𝐸, 𝐹), and that ‖𝑇‖ ≤ ‖𝑇‖𝑟 holds (see [95, Proposition 1.3.6]). We show that ‖⋅‖𝑟 is a Riesz norm on V(𝐸, 𝐹). Let 𝑆, 𝑇 ∈ V(𝐸, 𝐹) such that |𝑆| ≤ |𝑇|. Then |𝑆|𝑥 ≤ |𝑇|𝑥 for 𝑥 ∈ 𝐸+ , and therefore ‖|𝑆|𝑥‖ ≤ ‖|𝑇|𝑥‖ for 𝑥 ∈ 𝐸+ . Applying (4.1) to the positive operator |𝑆| yields ‖𝑆‖𝑟 = ‖|𝑆|‖ = sup{‖|𝑆|𝑥‖ : 𝑥 ∈ 𝐸+ , ‖𝑥‖ ≤ 1} ≤ sup{‖|𝑇|𝑥‖ : 𝑥 ∈ 𝐸+ , ‖𝑥‖ ≤ 1} = ‖|𝑇|‖ = ‖𝑇‖𝑟 . (2) Fix 𝑇 ∈ V(𝐸, 𝐹) and consider the set V0 = {𝑆 ∈ V+ (𝐸, 𝐹) : |𝑇𝑥| ≤ 𝑆|𝑥| for all 𝑥 ∈ 𝐸}.

Then |𝑇| ∈ V0 implies ‖|𝑇|‖ ≥ inf{‖𝑆‖ : 𝑆 ∈ V0 }, and so ‖𝑇‖𝑟 ≥ inf{‖𝑆‖ : 𝑆 ∈ V0 }. For each 𝑆 ∈ V0 one has |𝑇| ≤ 𝑆 by Proposition 4.24, i. e., |𝑇|𝑥 ≤ 𝑆𝑥, and so ‖|𝑇|𝑥‖ ≤ ‖𝑆𝑥‖, for 𝑥 ∈ 𝐸+ . This implies ‖|𝑇|‖ ≤ ‖𝑆‖ for all 𝑆 ∈ V0 , and finally, ‖𝑇‖𝑟 = ‖|𝑇|‖ ≤ inf{‖𝑆‖ : 𝑆 ∈ V0 }. Sufficient conditions for the norm completeness of (V(𝐸, 𝐹), ‖⋅‖𝑟 ) are provided in the next theorem, the proof of which is an adaptation of the well-known result for (L 𝑟 (𝐸, 𝐹), ‖⋅‖𝑟 ) if 𝐹 is Dedekind complete. If V(𝐸, 𝐹) is already known to be a vector lattice, then its norm completeness (with respect to the regular norm) can be proved by a similar argument, where, in general, the Dedekind completeness of 𝐹 is not required. Therefore the proof is provided in a little more detail. Theorem 4.26. Let 𝐸, 𝐹 be Banach lattices and V(𝐸, 𝐹) be a vector lattice of operators. If either (1) V(𝐸, 𝐹) = L 𝑟 (𝐸, 𝐹) or (2) V(𝐸, 𝐹) is closed with respect to the operator norm in L(𝐸, 𝐹), then (V(𝐸, 𝐹), ‖⋅‖𝑟 ) is a Banach lattice. Proof. The space (V(𝐸, 𝐹), ‖⋅‖𝑟 ) is a normed vector lattice by the previous theorem. Let (𝑇𝑛)𝑛∈ℕ be an arbitrary ‖⋅‖𝑟 - Cauchy sequence in V(𝐸, 𝐹). It should be shown that (𝑇𝑛)𝑛∈ℕ converges with respect to the regular norm to some operator 𝑇 ∈ V(𝐸, 𝐹). Since for the norms one has ‖⋅‖ ≤ ‖⋅‖𝑟 , it follows that ‖𝑇𝑚 − 𝑇𝑛 ‖ ≤ ‖𝑇𝑚 − 𝑇𝑛‖𝑟 for all 𝑛, 𝑚 ∈ ℕ, and so (𝑇𝑛)𝑛∈ℕ is a ‖⋅‖- Cauchy sequence as well, which converges in L(𝐸, 𝐹) to some operator 𝑇, i. e., ‖𝑇𝑛 − 𝑇‖ → 0. In the second case one automatically has 𝑇 ∈ V(𝐸, 𝐹). It will be shown that 𝑇 is a regular operator, and that ‖𝑇𝑛 −𝑇‖𝑟 → 0. Then in the first case one also has 𝑇 ∈ V(𝐸, 𝐹) = L 𝑟 (𝐸, 𝐹). Since (𝑇𝑛 )𝑛∈ℕ is a ‖⋅‖𝑟 - Cauchy sequence, there

86 | 4 Finite elements in vector lattices of linear operators is a subsequence (𝑇𝑚𝑛 ) − 𝑛 ∈ ℕ such that ‖𝑇𝑚𝑛 − 𝑇𝑚𝑛+1 ‖𝑟 ≤ 21𝑛 . Denote the sequence (𝑇𝑚𝑛 )𝑛∈ℕ again by (𝑇𝑛)𝑛∈ℕ , and define for each 𝑛 ∈ ℕ the operator ∞

𝑄𝑛 = ∑ |𝑇𝑘 − 𝑇𝑘+1 |, 𝑘=𝑛

where the convergence of the series in L(𝐸, 𝐹) with respect to the operator norm fol∞ 1 1 lows from ∑∞ 𝑘=𝑛 ‖|𝑇𝑘 − 𝑇𝑘+1 |‖ ≤ ∑𝑘=𝑛 2𝑘 = 2𝑛−1 . It is clear that 𝑄𝑛 ≥ 0, and 1 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑄𝑛󵄩󵄩 ≤ 𝑛−1 2

for 𝑛 ∈ ℕ.

(4.11)

For arbitrary fixed 𝑛 ∈ ℕ, any 𝑥 ∈ 𝐸 and 𝑚 ∈ ℕ with 𝑚 > 𝑛 the estimation ‖ |(𝑇 − 𝑇𝑛 )𝑥| − |(𝑇𝑚 − 𝑇𝑛)𝑥| ‖ ≤ ‖(𝑇 − 𝑇𝑛)𝑥 − (𝑇𝑚 − 𝑇𝑛 )𝑥‖ = ‖(𝑇 − 𝑇𝑚 )𝑥‖ ≤ ‖𝑇 − 𝑇𝑚 ‖‖𝑥‖ holds, and implies |(𝑇𝑚 − 𝑇𝑛 )𝑥| → |(𝑇 − 𝑇𝑛 )𝑥| as 𝑚 → ∞. It is clear that for 𝑚 > 𝑛, and 𝑥 ∈ 𝐸 one has 𝑚

𝑚

|(𝑇𝑚 − 𝑇𝑛)𝑥| ≤ ∑ |(𝑇𝑘 − 𝑇𝑘+1 )𝑥| ≤ ∑ |𝑇𝑘 − 𝑇𝑘+1 |(|𝑥|) ≤ 𝑄𝑛(|𝑥|), 𝑘=𝑛

𝑘=𝑛

which yields |(𝑇 − 𝑇𝑛 )𝑥| ≤ 𝑄𝑛(|𝑥|)

for 𝑥 ∈ 𝐸, 𝑛 ∈ ℕ.

(4.12)

This implies, in particular, 𝑇 − 𝑇𝑛 ≤ 𝑄𝑛, i. e., the regularity of 𝑇 − 𝑇𝑛, and because of 𝑇𝑛 ∈ V(𝐸, 𝐹) ⊆ L 𝑟 (𝐸, 𝐹), also the regularity of 𝑇. The operators 𝑄𝑛 belong to V(𝐸, 𝐹): in case (1) this follows from the positivity of 𝑄𝑛 and in case (2) it is clear, due to |𝑇𝑘 − 𝑇𝑘+1 | ∈ V(𝐸, 𝐹) for all 𝑘 ∈ ℕ. The Inequality (4.12) and Proposition 4.24 imply |𝑇 − 𝑇𝑛 | ≤ 𝑄𝑛. With (4.11) there holds the estimation 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩|𝑇 − 𝑇𝑛|𝑥󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑄𝑛𝑥󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑄𝑛󵄩󵄩󵄩 ‖𝑥‖ ≤

1 ‖𝑥‖ 2𝑛−1

for 𝑥 ∈ 𝐸+ , 𝑛 ∈ ℕ,

such that ‖𝑇 − 𝑇𝑛‖𝑟 = ‖|𝑇𝑛 − 𝑇|‖ = sup{‖|𝑇 − 𝑇𝑛 |𝑥‖ : 𝑥 ∈ 𝐸+ , ‖𝑥‖ ≤ 1} ≤

1 2𝑛−1

for all 𝑛 ∈ ℕ. It follows ‖𝑇 − 𝑇𝑛‖𝑟 → 0 as 𝑛 → ∞. Therefore, the constructed subsequence converges to 𝑇 ∈ V(𝐸, 𝐹) with respect to the regular norm, and so the original ‖⋅‖𝑟 - Cauchy sequence does. In general, since L 𝑟 (𝐸, 𝐹) may not be a vector lattice, the assumption (2) of the preceding theorem can not be replaced by the weaker condition (2󸀠 )

V(𝐸, 𝐹)

is closed with respect to the regular norm in L 𝑟 (𝐸, 𝐹).

However, the following is true.

4.4 Some vector lattices and Banach lattices of operators | 87

Remark 4.27. If L 𝑟 (𝐸, 𝐹) is a vector lattice then the condition (2󸀠 ) implies that V(𝐸, 𝐹) is a Banach lattice as well, if equipped with the norm ‖⋅‖𝑟 . We consider now some vector spaces of linear operators between Banach lattices 𝐸 and 𝐹. In several situations all operators of such a vector space possess their modulus and, consequently, this space turns out to be a vector lattice and the finite elements will be dealt with. In addition to the vector spaces of continuous and compact operators, we consider the following classes of operators between a Banach lattice 𝐸 and a Banach space 𝐹. Definition 4.28. Let be 𝐸 a Banach lattice and 𝑋, 𝑌 Banach spaces. – An operator 𝑇 : 𝐸 → 𝑌 is called 𝐴𝑀-compact, if for each 𝑎 ∈ 𝐸 the subset 𝑇 ({𝑥 ∈ 𝐸 : |𝑥| ≤ 𝑎}) is relatively compact in 𝑌 (see [43], [95, § 3.7]); – an operator 𝑇 : 𝑋 → 𝑌 is called weakly compact, if each bounded subset of 𝑋 is mapped onto a relatively weakly compact subset of 𝑌 (see [9, Sect.17], [41]); – an operator 𝑇 : 𝑋 → 𝑌 is called Dunford–Pettis (𝐷𝑃-operator), whenever 𝑥𝑛 → 0 󵄩 󵄩 weakly in 𝑋 implies 󵄩󵄩󵄩𝑇𝑥𝑛󵄩󵄩󵄩 → 0 (see [9, Sect.19]). Observe that all these (linear) operators are continuous. If 𝑌 is a Banach lattice, say 𝐹, denote the corresponding vector spaces by AM(𝐸, 𝐹), W(𝐸, 𝐹), and DP(𝐸, 𝐹) respectively. As before, by K(𝐸, 𝐹) we denote the ordered vector space of all compact operators. Then K(𝐸, 𝐹), K𝑟 (𝐸, 𝐹), AM(𝐸, 𝐹), W(𝐸, 𝐹), and DP(𝐸, 𝐹) are subspaces of L(𝐸, 𝐹) relevant for our further investigations. They are ordered vector spaces if equip them with the usual algebraic operations and if the order is introduced by means of the positive operators between 𝐸 and 𝐹. L(𝐸, 𝐹) L 𝑟 (𝐸, 𝐹) K(𝐸, 𝐹) K(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹) AM(𝐸, 𝐹) W(𝐸, 𝐹) DP(𝐸, 𝐹)

continuous operators regular operators compact operators compact regular operators 𝐴𝑀-compact operators weakly-compact operators Dunford-Pettis operators.

Special conditions on the Banach lattices 𝐸 and 𝐹 guarantee that the following ordered vector spaces of operators are vector lattices. Remark 4.29. (1) The spaces K(𝐸, 𝐹), where 𝐹 is an 𝐴𝑀-space, and K(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹), where 𝐸 is an 𝐴𝐿-space and 𝐹 has order continuous norm are both vector lattices; (Theorems of Krengel, [9, Theorems 16.7 and 16.9], see also [135, Theorem 3.1]). (2) The space K𝑟 (𝐸, 𝐹) = K+ (𝐸, 𝐹) − K+ (𝐸, 𝐹) is a 𝜎-Dedekind complete (Dedekind complete) vector lattice if one of the following three conditions holds: (i) both 𝐸󸀠 and 𝐹 have order continuous norm

88 | 4 Finite elements in vector lattices of linear operators (ii) 𝐸󸀠 is an atomic Banach lattice with order continuous norm and 𝐹 is 𝜎-Dedekind complete (Dedekind complete) (iii) 𝐹 is an atomic Banach lattice with order continuous norm, (Theorem of Wickstead, [135, Theorem 2.1 and Corollary 2.2]). (3) The space W(𝐸, 𝐹) is a vector lattice if 𝐸 is an 𝐴𝐿-space, and 𝐹 has the property (W1), or if 𝐸󸀠 has the property (W1), and 𝐹 is a Dedekind complete 𝐴𝑀-space with order unit; (Theorems of Chen and Wickstead, [41, Theorems 2.3 and 2.4]). (4) The spaces AM(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹), where 𝐹 has order continuous norm, and DP(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹), where 𝐸 has order continuous norm and 𝐹 is an 𝐴𝐿-space, are both vector lattices and even bands in L 𝑟 (𝐸, 𝐹); (Theorems of Dodds and Fremlin, [95, Theorems 3.7.2 and 3.7.20]). Observe that under the conditions in (1), the vector space K(𝐸, 𝐹) is a lattice-subspace of L 𝑟 (𝐸, 𝐹), but the latter need not be a vector lattice. Statement (3) holds if the Banach lattice 𝐹 is a 𝐾𝐵-space or 𝐸󸀠 is a 𝐾𝐵-space, since any 𝐾𝐵-space has property (W1); see [9, Corollary 17.15], and a Theorem of Schmidt in [110] and [95, Corollary 3.5.15]. Summing up the previous remarks, under the indicated conditions on the Banach lattices 𝐸 and 𝐹, the classes of operators (equipped with the natural algebraic and order operations) listed below constitute vector lattices: V(𝐸, 𝐹)

𝐸

𝐹

L (𝐸, 𝐹) K(𝐸, 𝐹) K(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹) W(𝐸, 𝐹) W(𝐸, 𝐹) AM(𝐸, 𝐹)∩L 𝑟 (𝐸, 𝐹) DP(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹)

arbitrary arbitrary 𝐴𝐿-space 𝐴𝐿-space 𝐸󸀠 is a 𝐾𝐵-space arbitrary order continuous norm

Dedekind complete 𝐴𝑀-space order continuous norm 𝐾𝐵-space Dedekind complete 𝐴𝑀-space with order unit order continuous norm 𝐴𝐿-space

𝑟

As a corollary of the previous theorem, we list some special cases where V(𝐸, 𝐹) is even a Banach lattice if it is equipped with the regular norm. Theorem 4.30. If 𝐸 and 𝐹 are Banach lattices, then (V(𝐸, 𝐹), ‖⋅‖𝑟 ) is a Banach lattice in the following situations: (a) V(𝐸, 𝐹) = L 𝑟 (𝐸, 𝐹), where 𝐹 is Dedekind complete; (b) V(𝐸, 𝐹) = L 𝑟 (𝐸, 𝐹), where 𝐸 is atomic with order continuous norm; (c) V(𝐸, 𝐹) = L(𝐸, 𝐹), where 𝐸 is an 𝐴𝐿-space and 𝐹 a 𝐾𝐵-space; (d) V(𝐸, 𝐹) = L(𝐸, 𝐹), where 𝐹 is a Dedekind complete with order unit; (e) V(𝐸, 𝐹) = L(𝐸, 𝐹), where 𝐸 is atomic with order continuous norm and 𝐹 has an order unit; (f) V(𝐸, 𝐹) = K(𝐸, 𝐹), where 𝐹 is an 𝐴𝑀-space;

4.4 Some vector lattices and Banach lattices of operators

| 89

(g) V(𝐸, 𝐹) = K(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹), where 𝐸 is an 𝐴𝐿-space and 𝐹 has order continuous norm; (h) V(𝐸, 𝐹) = W(𝐸, 𝐹), where 𝐸 is an 𝐴𝐿-space and 𝐹 has property (W1); (i) V(𝐸, 𝐹) = W(𝐸, 𝐹), where 𝐸󸀠 has property (W1) and 𝐹 is a Dedekind complete 𝐴𝑀space with order unit; (j) V(𝐸, 𝐹) = AM(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹), where 𝐹 has order continuous norm; (k) V(𝐸, 𝐹) = DP (𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹), where 𝐸 has order continuous norm and 𝐹 is an 𝐴𝐿-space. Proof. All listed spaces of operators are actually vector lattices as was mentioned in Section 4.4.1 and Remark 4.29. This allows us to apply Theorem 4.26 to the indicated vector lattices V(𝐸, 𝐹). The statements (a) and (b) are immediate by assumption (1) of the mentioned theorem. The statements (c), (d), and (e) follow from Theorem 4.26 (2). The statements (f), (h), and (i) also follow from Theorem 4.26 (2), since, according to [9, Theorems 16.1 and 17.4], the assumptions on 𝐸 and 𝐹 ensure that K(𝐸, 𝐹) and W(𝐸, 𝐹) are closed subspaces in the vector lattice L(𝐸, 𝐹) with respect to the operator norm. Statement (g) follows from Remark 4.27, since 𝑇𝑛 ∈ K(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹), 𝑇 ∈ 󵄩 󵄩 󵄩 󵄩 𝑟 L (𝐸, 𝐹) and 󵄩󵄩󵄩𝑇𝑛 − 𝑇󵄩󵄩󵄩𝑟 → 0 imply 󵄩󵄩󵄩𝑇𝑛 − 𝑇󵄩󵄩󵄩 → 0, and that 𝑇 is compact. Therefore 𝑇 ∈ V(𝐸, 𝐹). Finally, since the space 𝐹 is Dedekind complete in both statements (j) and (k), the space L 𝑟 (𝐸, 𝐹) under the regular norm is a Banach lattice according to statement (a). Therefore, in both of these cases, the space V(𝐸, 𝐹) is a band in L 𝑟 (𝐸, 𝐹), and so normclosed as every band. The statements now follow also from Remark 4.27. Corollary 4.31 ([2, Corollary 3.10]). If either 𝐸 is a Banach lattice and 𝐹 is a Dedekind complete Banach lattice with order unit, or 𝐸 is an 𝐴𝐿-space and 𝐹 is a 𝐾𝐵-space, then in both cases L(𝐸, 𝐹) is a Banach lattice with respect to the operator norm. Proof. Indeed, according to (d) of the preceding theorem, the space L(𝐸, 𝐹) coincides with L 𝑏 (𝐸, 𝐹) = L 𝑟 (𝐸, 𝐹) and is a Banach lattice with respect to the regular norm, where |𝑇|𝑥 is calculated by the Riesz–Kantorovich Formula (4.10) for 𝑇 ∈ L(𝐸, 𝐹), and 𝑥 ∈ 𝐸+ . Denote by 𝑢 > 0 an order unit in 𝐹 and let ‖⋅‖𝑢 be the corresponding order unit norm in 𝐹. Then ‖𝑢‖𝑢 = 1 and¹² 󵄩 󵄩 󵄩 󵄩 |𝑇𝑦| ≤ 󵄩󵄩󵄩𝑇𝑦󵄩󵄩󵄩𝑢 𝑢 ≤ ‖𝑇‖ 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 𝑢 ≤ ‖𝑇‖ ‖𝑥‖ 𝑢

for all |𝑦| ≤ 𝑥,

where 𝑥 ∈ 𝐸+ is fixed. The Formula (4.10) implies |𝑇|𝑥 ≤ ‖𝑇‖ ‖𝑥‖ 𝑢 and ‖|𝑇|𝑥‖𝑢 ≤ ‖𝑇‖ ‖𝑥‖ for 𝑥 ∈ 𝐸+ . This yields ‖𝑇‖𝑟 = sup{‖|𝑇|𝑥‖𝑢 : ‖𝑥‖ ≤ 1, 𝑥 ∈ 𝐸+ } ≤ ‖𝑇‖. The inverse

12 The operator norm for 𝑇 ∈ L(𝐸, 𝐹), with respect to the order unit norm in 𝐹, is also denoted by ‖𝑇‖.

90 | 4 Finite elements in vector lattices of linear operators inequality ‖𝑇‖ ≤ ‖𝑇‖𝑟 is always true. The second case, when 𝐸 is an 𝐴𝐿-space and 𝐹 is a 𝐾𝐵-space, follows from (c) of the previous theorem and [2, Corollary 3.10], where the equality ‖𝑇‖ = ‖𝑇‖𝑟 is proved under the formulated conditions, i. e., the operator norm and the regular norm for 𝑇 ∈ L(𝐸, 𝐹) coincide. Several other situations are known where some classes of linear continuous operators on Banach lattices form a vector lattice, or even a normed or a Banach lattice with respect to an appropriate norm (operator norm, regular or another norm). See e. g., [35, Theorem 2.2], [2, Theorems 3.14(b) and 3.12(2)], [111, Corollary III.4.14], [41, § 3,], [32, 53]. For example, in [135, Theorem 3.1], there are described the only four cases of combinations of Banach lattices 𝐸 and 𝐹 for K(𝐸, 𝐹) to be a 𝜎-Dedekind complete or a Dedekind complete Banach lattice under the operator norm.

4.5 Operators as finite elements Our further aim is the study of finite elements in vector lattices of linear operators, where we are interested in vector lattices V(𝐸, 𝐹) with Φ1 (V(𝐸, 𝐹)) = V(𝐸, 𝐹). For orthomorphisms in L 𝑟 (𝐸, 𝐹) we already have the following results: – if 𝐸 is a Banach lattice, then (Orth(𝐸), ‖⋅‖𝑟 ) is an 𝐴𝑀-space with order unit and, by Theorem 3.6, Orth(𝐸) = Φ1 (Orth(𝐸)); – if 𝐸 is a Dedekind complete Banach lattice, then all orthomorphisms on 𝐸 are totally finite elements in the vector lattice of all regular operators (Theorem 4.1). The Dedekind completeness of 𝐸 in the last result can be dropped if L 𝑟 (𝐸) is known to be a vector lattice¹³, as can be seen from the next theorem. Theorem 4.32. If 𝐸 is a Banach lattice and L 𝑟 (𝐸) is a vector lattice, then each 𝑇 ∈ Orth(𝐸) is a finite element in L 𝑟 (𝐸), i. e., Orth(𝐸) ⊂ Φ1 (L 𝑟 (𝐸)). Proof. Since 𝐸 is a Banach lattice, by Theorem 4.26 (L 𝑟 (𝐸), ‖⋅‖𝑟 ) is a Banach lattice as well and, by the cited Wickstead’s Theorem (see [9, Theorem 15.5]), one has Orth(𝐸) = {𝑇 ∈ L 𝑏 (𝐸) : ∃ 𝜆 > 0 with − 𝜆𝐼 ≤ 𝑇 ≤ 𝜆𝐼}. Therefore Orth(𝐸) = {𝑇 ∈ L 𝑟 (𝐸) : ∃ 𝜆 > 0 with |𝑇| ≤ 𝜆𝐼} is the ideal generated by 𝐼 in L 𝑟 (𝐸). It is even closed with respect to the regular norm¹⁴. In view of Theorem 3.6, one has Orth(𝐸) = Φ1 (Orth(𝐸)), and the proof of the theorem is completed by applying Theorem 3.24, which gives Φ1 (Orth(𝐸)) ⊂ Φ1 (L 𝑟 (𝐸)). Of course, as a special case of this theorem we get the part Orth(𝐸) ⊆ Φ1 (L 𝑟 (𝐸)) from Theorem 4.1. In the proof of that theorem, the Dedekind completeness of 𝐸 was es-

13 Of course, in this case L 𝑟 (𝐸) ≠ L 𝑏 (𝐸), see Theorem (2.5). 14 This can easily be proved directly.

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sentially used not only in order to ensure that L 𝑟 (𝐸) is a vector lattice, but also for the principal projection property to be held in L 𝑟 (𝐸) which allowed the application of Theorem 3.13. Corollary 4.33. If 𝐸 is a Dedekind complete Banach lattice, then Orth(𝐸) ⊆ Φ1 (L 𝑟 (𝐸)). In particular, the identity operator on 𝐸 is a finite element in L 𝑟 (𝐸). A general result for appropriate V(𝐸, 𝐹), similar to L 𝑟 (𝐸, 𝐹), in Theorem 4.10, is established below. Theorem 4.34. Let 𝐸 be an 𝐴𝐿-space and 𝐹 a Banach lattice with order unit 𝑢 ∈ 𝐹+ . Let V(𝐸, 𝐹) be a vector lattice of operators which contains all rank one operators of the kind 𝑓 ⊗ 𝑢 for 𝑓 ∈ 𝐸󸀠 . Then V(𝐸, 𝐹) possesses a rank one operator as an order unit, and therefore each 𝑇 ∈ V(𝐸, 𝐹) is a finite element of V(𝐸, 𝐹), i. e., Φ𝑖 (V(𝐸, 𝐹)) = V(𝐸, 𝐹) for 𝑖 = 1, 2, 3. Proof. If the norm in 𝐹 is replaced by the (equivalent) order unit norm ‖⋅‖𝑢 , then |𝑦| ≤ 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑦󵄩󵄩𝑢 𝑢 for each 𝑦 ∈ 𝐹. Therefore ±𝑇𝑥 ≤ |𝑇𝑥| ≤ ‖𝑇𝑥‖𝑢 𝑢 ≤ ‖𝑇‖ ‖𝑥‖ 𝑢

for all 𝑥 ∈ 𝐸.

The positive linear functional 𝑓0 defined on the 𝐴𝐿-space 𝐸 by 𝑓0 (𝑥) = ‖𝑥+ ‖−‖𝑥− ‖ is an order unit in 𝐸󸀠 (see [2, Theorem 3.3.]), where ±𝑇𝑥 ≤ ‖𝑇‖ 𝑓0 (𝑥)𝑢 holds for all 𝑥 ∈ 𝐸+ . The last inequality can be written as ±𝑇𝑥 ≤ ‖𝑇‖ (𝑓0 ⊗ 𝑢)(𝑥) for 𝑥 ∈ 𝐸+ . Since by assumption the operator 𝑈 = 𝑓0 ⊗ 𝑢 belongs to the vector lattice V(𝐸, 𝐹), one has |𝑇| ≤ ‖𝑇‖𝑈 for any 𝑇 ∈ V(𝐸, 𝐹), i. e., 𝑈 is an order unit in V(𝐸, 𝐹). In order to complete the proof we refer to Proposition 3.44. Since the rank one operators obviously belong to each of the spaces L 𝑟 (𝐸, 𝐹), K(𝐸, 𝐹), and W(𝐸, 𝐹), the following corollaries are simple consequences of the proved theorem. Corollary 4.35 ([38, Theorem 8(a)]). Let 𝐸 be an 𝐴𝐿-space and 𝐹 a Dedekind complete Banach lattice with order unit. Then Φ𝑖 (L 𝑟 (𝐸, 𝐹)) = L 𝑟 (𝐸, 𝐹) = L(𝐸, 𝐹) for 𝑖 = 1, 2, 3. Indeed, L 𝑟 (𝐸, 𝐹) is a vector lattice since 𝐹 is Dedekind complete and L 𝑟 (𝐸, 𝐹) = L(𝐸, 𝐹) holds; see [6, 138]. Corollary 4.36. Let 𝐸 be an atomic 𝐴𝐿-space and 𝐹 a Banach lattice with order unit. Then Φ𝑖 (L 𝑟 (𝐸, 𝐹)) = L 𝑟 (𝐸, 𝐹) for 𝑖 = 1, 2, 3. Since an 𝐴𝐿-space has order continuous norm, the space L 𝑏 (𝐸, 𝐹) is a vector lattice and L 𝑏 (𝐸, 𝐹) = L 𝑟 (𝐸, 𝐹). So L 𝑟 (𝐸, 𝐹) is a vector lattice and the theorem can be applied. Corollary 4.37. Let 𝐸 be an 𝐴𝐿-space and 𝐹 an 𝐴𝑀-space with order unit. Then Φ𝑖 (K(𝐸, 𝐹)) = K(𝐸, 𝐹) for 𝑖 = 1, 2, 3.

92 | 4 Finite elements in vector lattices of linear operators Indeed, due to Theorem 4.30 (f), the space K(𝐸, 𝐹) is a vector lattice and the theorem can be applied. Corollary 4.37 implies, in particular, that any compact operator 𝑇 : 𝐿 1 [0, 1] → 𝐶[0, 1] is a finite element in K(𝐿 1 [0, 1], 𝐶[0, 1]). Corollary 4.38. If 𝐹 is a 𝐾𝐵-space 𝐹 with order unit then dim 𝐹 < ∞ (see [122, Theorem 1.2.4]). Then the case (ℎ) of Theorem 4.30, in combination with the last theorem, shows that the vector lattice W(𝐸, 𝐹) = L(𝐸, 𝐹) coincides with the vector space F(𝐸, 𝐹) of all finite rank operators between 𝐸 and 𝐹. So F(𝐸, 𝐹) is a vector lattice and Φ𝑖 (F(𝐸, 𝐹)) = F(𝐸, 𝐹) for 𝑖 = 1, 2, 3 by Corollary 4.35. Remark 4.39. Under the assumptions of Theorem 4.34 , the order unit 𝑈 = 𝑓0 ⊗ 𝑢 can be used to equip the vector lattice V(𝐸, 𝐹) with the corresponding order unit norm ‖⋅‖𝑈 . If it is known that V(𝐸, 𝐹) (under some norm) is a Banach lattice, then (V(𝐸, 𝐹), ‖⋅‖𝑈 ) turns out to be an 𝐴𝑀-space with the order interval [−𝑈, 𝑈] as its closed unit ball. Then (V(𝐸, 𝐹), ‖⋅‖𝑟 ) is an 𝐴𝑀-space with order unit and, in particular, ‖𝑇‖𝑟 = ‖𝑇‖𝑈 . Indeed, let 𝐹 be equipped with the equivalent order unit norm, which is denoted by ‖⋅‖. Since 𝑈 = 𝑓0 ⊗ 𝑢 is a positive operator and ‖𝑢‖ = 1, by Theorem 4.25 one has ‖𝑈‖𝑟 = ‖𝑈‖ = sup{‖𝑓0 (𝑥)𝑢‖ : ‖𝑥‖ ≤ 1, 𝑥 ∈ 𝐸+ } = sup{‖𝑥‖ : ‖𝑥‖ ≤ 1} = 1. On the other hand, since V(𝐸, 𝐹) is assumed to be a Banach lattice, (V(𝐸, 𝐹), ‖⋅‖𝑈 ) is an 𝐴𝑀-space, where ‖𝑈‖𝑈 = 1 and |𝑇| ≤ ‖𝑇‖𝑈 𝑈 for 𝑇 ∈ V(𝐸, 𝐹). Since ‖⋅‖𝑟 is a Riesz norm, the last relation implies ‖𝑇‖𝑟 ≤ ‖𝑇‖𝑈 ‖𝑈‖𝑟 = ‖𝑇‖𝑈 . Vice versa, |𝑇| ≤ ‖𝑇‖𝑈 ≤ ‖𝑇‖𝑟 𝑈, as has been shown in the proof of the theorem, implies ‖𝑇‖𝑈 ≤ ‖𝑇‖𝑟 ‖𝑈‖𝑈 = ‖𝑇‖𝑟 . Thus ‖𝑇‖𝑈 = ‖𝑇‖𝑟 , and so (V(𝐸, 𝐹), ‖ ⋅ ‖𝑟 ) is an 𝐴𝑀space with order unit.

4.6 Finite rank operators as finite elements If, for Banach lattices 𝐸, 𝐹, and some Banach lattice of operators V(𝐸, 𝐹) containing all rank one operators, either 𝐸 is not lattice isomorphic to an 𝐴𝑀-space, or 𝐹 is not lattice isomorphic to an 𝐴𝐿-space, then, in general, Φ1 (V(𝐸, 𝐹)) ≠ V(𝐸, 𝐹). The next result is similar to Theorem 4.14, where the case V(𝐸, 𝐹) = L 𝑟 (𝐸, 𝐹) for Dedekind complete 𝐹 was considered. In this section we are able to extend some results which were proved in Section 4.4 for the vector lattice L 𝑟 (𝐸, 𝐹) under the assumption of Dedekind completeness of 𝐹, to other vector lattices of operators. Theorem 4.40. Let 𝐸 and 𝐹 be Banach lattices, and V(𝐸, 𝐹) be a vector lattice of operators which contains all rank one operators. If, for nonzero elements 𝑓 ∈ 𝐸󸀠 and 𝑣 ∈ 𝐹, the rank one operator 𝑓 ⊗ 𝑣 is a finite element in V(𝐸, 𝐹), then 𝑓 is finite in 𝐸󸀠 and 𝑣 is finite in 𝐹.

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Proof. Some operator 𝑈 ∈ V(𝐸, 𝐹) exists, such that for each 𝑆 ∈ V(𝐸, 𝐹) and some 𝑐𝑆 > 0, one has |𝑆| ∧ 𝑛|𝑓 ⊗ 𝑣| ≤ 𝑐𝑆 𝑈 for all 𝑛 ∈ ℕ. (4.13) For arbitrary 𝑦 ∈ 𝐹, the operator 𝑆 = 𝑓 ⊗ 𝑦 belongs to V(𝐸, 𝐹) and |𝑆| = |𝑓| ⊗ |𝑦|. Then, by the last inequality, one obtains |𝑓| ⊗ (|𝑦| ∧ 𝑛|𝑣|) = (|𝑓| ⊗ |𝑦|) ∧ (|𝑓| ⊗ 𝑛|𝑣|) = |𝑆| ∧ (|𝑓| ⊗ 𝑛|𝑣|) ≤ 𝑐𝑆 𝑈,

𝑛 ∈ ℕ.

So, for all 𝑥 ∈ 𝐸+ one has |𝑓(𝑥)|(|𝑦| ∧ 𝑛|𝑣|) ≤ 𝑐𝑆 𝑈(𝑥) which, in particular, is true for some 𝑥0 ∈ 𝐸+ with |𝑓(𝑥0 )| = 1. Then |𝑦| ∧ 𝑛|𝑣| ≤ 𝑐𝑆 𝑈(𝑥0 ) shows that the element 𝑣 is finite in 𝐹 with 𝑈(𝑥0 ) as one of its 𝐹-majorants. The finiteness of the element 𝑓 is established analogously. This time the inequality (4.13) with the operator 𝑆 = 𝑔 ⊗ 𝑣 for 𝑔 ∈ 𝐸󸀠 gives (|𝑔| ∧ 𝑛|𝑓|) ⊗ |𝑣| = |𝑔 ⊗ 𝑣| ∧ 𝑛|𝑓 ⊗ 𝑣| ≤ 𝑐𝑆 𝑈,

𝑛 ∈ ℕ,

which yields (|𝑔| ∧ 𝑛|𝑓|) (𝑥)|𝑣| ≤ 𝑐𝑆 𝑈(𝑥) for 𝑥 ∈ 𝐸+ . If 𝑧󸀠 ∈ 𝐹+󸀠 , then (|𝑔| ∧ 𝑛|𝑓|) (𝑥)𝑧󸀠 (|𝑣|) ≤ 𝑐𝑆 𝑧󸀠 (𝑈(𝑥)),

𝑥 ∈ 𝐸+ , 𝑛 ∈ ℕ.

If now 𝑈󸀠 : 𝐹󸀠 → 𝐸󸀠 denotes the adjoint operator to 𝑈, and 𝑧0󸀠 ∈ 𝐹+󸀠 satisfies 𝑧0󸀠 (|𝑣|) = 1, then (|𝑔| ∧ 𝑛|𝑓|) (𝑥) ≤ 𝑐𝑆 (𝑈󸀠 𝑧0󸀠 )(𝑥) for 𝑥 ∈ 𝐸+ , 𝑛 ∈ ℕ. With 𝑈󸀠 𝑧0󸀠 = 𝑓0 there follows (|𝑔| ∧ 𝑛|𝑓|) ≤ 𝑐𝑆 𝑓0 , i. e., the element 𝑓 is finite in 𝐸󸀠 with 𝑈󸀠 𝑧0󸀠 as one of its 𝐸󸀠 -majorant. In Theorem 4.14, as already mentioned before Theorem 4.32, the Dedekind completeness of 𝐹 was not only responsible for L 𝑟 (𝐸, 𝐹) being a vector lattice, but enabled us to prove also the inverse statement that 𝑓 ∈ Φ1 (𝐸󸀠 ) and 𝑣 ∈ Φ1 (𝐹) imply the finiteness of the operator 𝑓 ⊗ 𝑣 in L 𝑟 (𝐸, 𝐹). The idea of that proof was based on two facts: that {𝑣}⊥⊥ is a projection band, and that it possesses an order unit as far as the vector 𝑣 is assumed to be a finite element in 𝐹 (see also Remark 3.16 (2)). In general, the element 𝑓⊗𝑢 may not be finite in the corresponding ambient vector lattice of operators even if the latter is a Banach lattice. This will be demonstrated in the two situations described next. Let 𝐸 be an atomic Banach lattice with order continuous norm, and 𝐹 an arbitrary Banach lattice (not necessarily Dedekind complete) which is not lattice isomorphic to an 𝐴𝑀-space. Then according to Theorem 3.19, one has Φ1 (𝐹) ≠ 𝐹. So there is some 0 ≠ 𝑣 ∈ 𝐹 ∖ Φ1 (𝐹), such that by the previous theorem the operator 𝑓 ⊗ 𝑣 is not a finite element in L 𝑟 (𝐸, 𝐹), whenever 0 ≠ 𝑓 ∈ 𝐸󸀠 , although L 𝑟 (𝐸, 𝐹) is a Banach lattice by Theorem 4.30 (b). Similarly, if 𝐸 is a Banach lattice which is not lattice isomorphic to an 𝐴𝐿-space, and 𝐹 is an 𝐴𝑀-space, then K(𝐸, 𝐹) is a Banach lattice by Theorem 4.30 (f). Since now 𝐸󸀠 is not lattice isomorphic to an 𝐴𝑀-space, again by Theorem 3.19 there is some

94 | 4 Finite elements in vector lattices of linear operators 0 ≠ 𝑓 ∈ 𝐸󸀠 ∖ Φ1 (𝐸󸀠 ) such that the operator 𝑓 ⊗ 𝑢 does not belong to Φ1 (K(𝐸, 𝐹)), whenever 0 ≠ 𝑢 ∈ 𝐹. So the finiteness of both elements 𝑓 and 𝑢 in their respective vector lattices 𝐸󸀠 and 𝐹 is a necessary condition for the finiteness of 𝑓⊗𝑢 in the corresponding operator lattice, and one has to look for sufficient conditions. The finiteness of finite rank operators in L 𝑟 (𝐸, 𝐹) was described in Section 4.3. Based on Corollary 4.15 and on Theorem 4.30 it is now possible to recognize finite rank operators with finite constituents¹⁵ as finite elements in some other vector lattices V(𝐸, 𝐹). Theorem 4.41. Let 𝐸, 𝐹 be Banach lattices, and V(𝐸, 𝐹) be an arbitrary vector lattice of operators. Let 𝑓1 , . . . , 𝑓𝑛 be finite elements in 𝐸󸀠 , and 𝑣1 , . . . , 𝑣𝑛 finite elements in 𝐹. Then the operator 𝑇 = ∑𝑛𝑖=1 𝑓𝑖 ⊗ 𝑣𝑖 is a finite element in V(𝐸, 𝐹) in the following situations: (a) 𝐸 is an 𝐴𝐿-space, 𝐹 has order continuous norm and V(𝐸, 𝐹) = K(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹); (b) 𝐸 is an 𝐴𝐿-space, 𝐹 is a 𝐾𝐵-space, and V(𝐸, 𝐹) = W(𝐸, 𝐹); (c) 𝐸󸀠 is a 𝐾𝐵-space, 𝐹 is a Dedekind complete 𝐴𝑀-space with order unit, and V(𝐸, 𝐹) = W(𝐸, 𝐹); (d) 𝐹 has order continuous norm and V(𝐸, 𝐹) = AM(𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹); (e) 𝐸 has order continuous norm, 𝐹 is an 𝐴𝐿-space, and V(𝐸, 𝐹) = DP (𝐸, 𝐹) ∩ L 𝑟 (𝐸, 𝐹). Proof. From Theorem 4.30 (g)–(k) we know that in each of the indicated cases the space V(𝐸, 𝐹) is a vector lattice. Since any rank one operator is compact and any compact operator is weakly compact, 𝐴𝑀-compact, and also a 𝐷𝑃-operator, the operator 𝑇 belongs to V(𝐸, 𝐹) in every case (a)–(e). Notice that the space 𝐹 throughout the theorem is Dedekind complete. By Corollary 4.15, the operator 𝑇 is a finite element in L 𝑟 (𝐸, 𝐹) with a rank one operator as one of its majorants. Since such a majorant (as a compact operator) belongs to V(𝐸, 𝐹), the operator 𝑇 is also a finite element in V(𝐸, 𝐹). It is clear that the statements of Theorem 4.41 give certain information only in the case Φ1 (𝐸󸀠 ) ≠ {0}. E. g., 𝐸 = 𝐿 𝑝 (0, 1) with 𝑝 ∈ (1, ∞) is excluded due to Φ1 (𝐸󸀠 ) = {0} (the latter was mentioned in (b) after Theorem 3.18). On the other hand, if 𝐸 is an 𝐴𝐿-space, then 𝐸󸀠 is an 𝐴𝑀-space with order unit, and all elements of 𝐸󸀠 are finite. Theorem 4.42. Let 𝐸 be an 𝐴𝐿-space, 𝐹 be a Banach lattice, and let 𝐵 be a band in 𝐹. Let V(𝐸, 𝐹) be a Banach lattice of operators which contains all rank one operators such that for the modulus |𝑇| of each operator 𝑇 ∈ V(𝐸, 𝐹), the Riesz-Kantorovich formulae (4.10) hold. Then (1) V(𝐸, 𝐵) = {𝑇 ∈ V(𝐸, 𝐹) : 𝑇(𝐸) ⊂ 𝐵} is a closed ideal of V(𝐸, 𝐹); (2) if the band 𝐵 has an order unit, then V(𝐸, 𝐵) = Φ1 (V(𝐸, 𝐵)), and V(𝐸, 𝐵) ⊂ Φ1 (V(𝐸, 𝐹)).

15 I. e., 𝑓𝑖 ∈ Φ1 (𝐸󸀠 ), and 𝑦𝑖 ∈ Φ1 (𝐹) for 𝑖 ∈ ℕ in some representation of the operator 𝑇 = ∑𝑛𝑖=1 𝑓𝑖 ⊗ 𝑦𝑙 .

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Proof. (1) Obviously V(𝐸, 𝐵) is a subspace of V(𝐸, 𝐹). For 𝑇 ∈ V(𝐸, 𝐵), the vector |𝑇𝑦| belongs to 𝐵 for any |𝑦| ≤ 𝑥, where 𝑥 ∈ 𝐸+ . This implies sup{|𝑇𝑦| : |𝑦| ≤ 𝑥} ∈ 𝐵, and the last expression is equal to |𝑇|𝑥 by the Formula (4.10). Then |𝑇| belongs to V(𝐸, 𝐵) for each 𝑇 ∈ V(𝐸, 𝐵), which shows that V(𝐸, 𝐵) is a vector sublattice of V(𝐸, 𝐹). If 𝑆 ∈ V(𝐸, 𝐹), 𝑇 ∈ V(𝐸, 𝐵) are such that 0 ≤ 𝑆 ≤ 𝑇, then 0 ≤ 𝑆𝑥 ≤ 𝑇𝑥 for 𝑥 ∈ 𝐸+ implies 𝑆𝑥 ∈ 𝐵, and so 𝑆 ∈ V(𝐸, 𝐵). The proof of the closedness of V(𝐸, 𝐵) in V(𝐸, 𝐹) uses the closedness of 𝐵 in 𝐹 and is standard. (2) Let 𝑢 > 0 be an order unit in the band 𝐵. Then V(𝐸, 𝐵) is a vector lattice which contains all rank one operators of the kind 𝑓 ⊗ 𝑢 for 𝑓 ∈ 𝐸󸀠 . Since 𝐵 with the induced norm of 𝐹 is closed ([95, Proposition 1.2.3]), it is a Banach lattice. According to Theorem 4.34, one now has V(𝐸, 𝐵) = Φ1 (V(𝐸, 𝐵)). By the first part of the proof, V(𝐸, 𝐵) is a closed ideal in V(𝐸, 𝐹), and therefore by Theorem 3.24, one has Φ1 (V(𝐸, 𝐵)) ⊂ Φ1 (V(𝐸, 𝐹)). Concerning the finiteness of finite rank operators between two Banach lattices 𝐸 and 𝐹, most results are known if 𝐹 is Dedekind complete (cf. Section 4.3). We now consider finite rank operators in K(𝐸, 𝐹), where 𝐹 is assumed to be an 𝐴𝑀-space with order unit. Then it is clear that the ordered vector space K(𝐸, 𝐹) is a Banach lattice (cf. Theorem 4.30 (f)). Theorem 4.43. Let 𝐸 be a Banach lattice and 𝐹 an 𝐴𝑀-space with order unit. If 𝜑 ∈ Φ1 (𝐸󸀠 ), then for any 𝑦 ∈ 𝐹 the rank one operator 𝜑 ⊗ 𝑦 is a finite element in K(𝐸, 𝐹). Proof. Without loss of generality, the operator 𝑇 = 𝜑 ⊗ 𝑦 is assumed to be positive. Denote the order unit of 𝐹 by 𝑢0 and assume 𝑢0 > 0. The space 𝐸󸀠 is Dedekind complete and 𝜑 is finite in 𝐸󸀠 . So, by Theorem 3.11, the band 𝐵󸀠 = {𝜑}𝑑𝑑 has an order unit 0 < 𝑓0 ∈ 𝐵󸀠 . We show that the operator 𝑈0 = 𝑓0 ⊗ 𝑢0 ∈ K(𝐸, 𝐹) is a majorant for 𝑇. Let 𝑆 ∈ K(𝐸, 𝐹) be arbitrary. If 𝑇󸀠 denotes the adjoint operator to 𝑇, then for any 𝑢󸀠 ∈ 𝐹+󸀠 one has 𝑇󸀠 𝑢󸀠 = 𝑢󸀠 (𝑦)𝜑 = (𝑢󸀠 ⊗ 𝜑)(𝑦) ∈ 𝐵󸀠 , and therefore, (|𝑆| ∧ 𝑛𝑇)󸀠 𝑢󸀠 ≤ 𝑛𝑇󸀠 𝑢󸀠 ∈ 𝐵󸀠 implies (|𝑆| ∧ 𝑛𝑇)󸀠 𝑢󸀠 ∈ 𝐵󸀠 for all 𝑛 ∈ ℕ and 𝑢󸀠 ∈ 𝐹+󸀠 . 𝐵󸀠 as a closed subspace of 𝐸󸀠 is a Banach lattice in its own. If 𝐵󸀠 is equipped with 󵄩 󵄩 the norm generated by the order unit 𝑓0 , then |𝑓| ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩 𝑓0 for 𝑓 ∈ 𝐵󸀠 . In particular, 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 (|𝑆| ∧ 𝑛𝑇)󸀠 𝑢󸀠 ≤ 󵄩󵄩󵄩󵄩(|𝑆| ∧ 𝑛𝑇)󸀠 𝑢󸀠 󵄩󵄩󵄩󵄩 𝑓0 ≤ 󵄩󵄩󵄩󵄩(|𝑆| ∧ 𝑛𝑇)󸀠 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝑢󸀠 󵄩󵄩󵄩󵄩 𝑓0 = ‖|𝑆| ∧ 𝑛𝑇‖ 𝑢󸀠 (𝑢0 )𝑓0 ≤ ‖|𝑆| ∧ 𝑛𝑇‖𝑟 𝑢󸀠 (𝑢0 )𝑓0 ≤ ‖𝑆‖𝑟 𝑢󸀠 (𝑢0 )𝑓0 for 𝑛 ∈ ℕ and 𝑢󸀠 ∈ 𝐹+󸀠 . For 𝑥 ∈ 𝐸+ this means ((|𝑆|∧𝑛𝑇)󸀠 𝑢󸀠 )(𝑥) ≤ ‖𝑆‖𝑟 𝑢󸀠 (𝑢0 )𝑓0 (𝑥), which can be written as 𝑢󸀠 ((|𝑆| ∧ 𝑛𝑇)(𝑥)) = ((|𝑆| ∧ 𝑛𝑇)󸀠 𝑢󸀠 )(𝑥) ≤ ‖𝑆‖𝑟 𝑢󸀠 (𝑢0 )𝑓0 (𝑥) = ‖𝑆‖𝑟 𝑢󸀠 (𝑈0 (𝑥)) for 𝑛 ∈ ℕ, 𝑥 ∈ 𝐸+ and all 𝑢󸀠 ∈ 𝐹+󸀠 . Therefore, (|𝑆| ∧ 𝑛𝑇)(𝑥) ≤ ‖𝑆‖𝑟 𝑈0 (𝑥) for all 𝑥 ∈ 𝐸+ and 𝑛 ∈ ℕ. Finally, one has |𝑆| ∧ 𝑛𝑇 ≤ ‖𝑆‖𝑟 𝑈0 for all 𝑛 ∈ ℕ, i. e., 𝑇 is finite in K(𝐸, 𝐹) with 𝑈0 = 𝑓0 ⊗ 𝑢0 as one of its majorants.

96 | 4 Finite elements in vector lattices of linear operators Together with Theorem 4.40, one immediately has the following corollary. Corollary 4.44. Let 𝐸 be a Banach lattice and 𝐹 an 𝐴𝑀-space with order unit. If 𝜑 ∈ 𝐸󸀠 and 0 ≠ 𝑦 ∈ 𝐹, then the operator 𝜑 ⊗ 𝑦 is finite in K(𝐸, 𝐹) if and only if 𝜑 ∈ Φ1 (𝐸󸀠 ). Corollary 4.45. Let 𝐸 be a Banach lattice and 𝐹 an 𝐴𝑀-space with order unit. Let 𝜑1 , . . . , 𝜑𝑛 ∈ Φ1 (𝐸󸀠 ) and 𝑦1 , . . . , 𝑦𝑛 ∈ 𝐹. Then the operator ∑𝑛𝑖=1 𝜑𝑖 ⊗ 𝑦𝑖 is finite in K(𝐸, 𝐹). Indeed, by the theorem the operator 𝜑𝑖 ⊗ 𝑦𝑖 is finite in K(𝐸, 𝐹) for each 𝑖 = 1, . . . , 𝑛. Then so is ∑𝑛𝑖=1 𝜑𝑖 ⊗ 𝑦𝑖 .

4.7 Impact of the order structure of V(𝐸, 𝐹) on the lattice properties of 𝐸 and 𝐹 If in Theorem 4.34 the vector lattice (V(𝐸, 𝐹), ‖⋅‖𝑟 ) is assumed to be a Banach lattice (e. g., if condition (1) of Theorem 4.26 holds), then according to Remark 4.39, the space V(𝐸, 𝐹) is an 𝐴𝑀-space. The inverse statement is also true. It slightly generalizes a well-known result of Wickstead. For its formulation we need the following notion: the norm in a Banach lattice 𝐸 has the Fatou-property¹⁶ , or equivalently, is said to satisfy the condition (C) (see [59, Chap. X.4]), if 0 ≤ 𝑥𝛼 ↑ 𝑥 ∈ 𝐸 implies sup ‖𝑥𝛼 ‖ = ‖𝑥‖. The norm is sequentially Fatou if the last holds only for sequences in 𝐸. The theorem cited is then the following. Theorem 4.46 ([136, Theorem 2.2]). For two Banach lattices 𝐸 and 𝐹, such that the order intervals in 𝐸 are separable and 𝐹 is 𝜎-Dedekind complete, the space (L 𝑟 (𝐸, 𝐹), ‖⋅‖𝑟 ) is an 𝐴𝑀-space if and only if the following statements hold: (a) 𝐸 is an 𝐴𝐿-space; (b) 𝐹 is an 𝐴𝑀-space; (c) at least one of the conditions holds: (c1 ) 𝐸 is atomic, or (c2 ) the norm in 𝐹 is a sequentially Fatou norm. Further on in this section, the normed vector lattice V(𝐸, 𝐹) is always assumed to be equipped with the regular norm. Theorem 4.47. Let 𝐸 and 𝐹 be Banach lattices. If V(𝐸, 𝐹) is an 𝐴𝑀-space with respect to the regular norm and contains all rank one operators, then 𝐸 is an 𝐴𝐿-space and 𝐹 an 𝐴𝑀-space. Proof. Let 𝑓 ∈ 𝐸󸀠+ and 𝑦 ∈ 𝐹+ . Then the norm of the rank one operator 𝑓 ⊗ 𝑦 : 𝐸 → 𝐹 calculates as ‖𝑓 ⊗ 𝑦‖𝑟 = sup{‖𝑓(𝑥)𝑦‖ : ‖𝑥‖ ≤ 1} = ‖𝑦‖‖𝑓‖. According to Proposition 2.10, for 𝐸 to be an 𝐴𝐿-space, it is sufficient to show that 𝐸󸀠 is an 𝐴𝑀-space. For 𝑦 ∈ 𝐹+ , such that ‖𝑦‖ = 1 and 𝑓1 , 𝑓2 ∈ 𝐸󸀠+ , the rank one operators 𝑇𝑖 = 𝑓𝑖 ⊗ 𝑦, (𝑖 = 1, 2) belong to V(𝐸, 𝐹). Then by using V(𝐸, 𝐹) as an 𝐴𝑀-

16 In this case ‖⋅‖ is said to be a Fatou norm.

4.7 Impact of the order structure of V(𝐸, 𝐹) on the lattice properties of 𝐸 and 𝐹

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space, one has ‖𝑓1 ∨ 𝑓2 ‖ = ‖(𝑓1 ∨ 𝑓2 ) ⊗ 𝑦‖𝑟 = ‖(𝑓1 ⊗ 𝑦) ∨ (𝑓2 ⊗ 𝑦)‖𝑟 = max{‖𝑇1 ‖𝑟 , ‖𝑇2 ‖𝑟 } = max{‖𝑓1 ‖‖𝑓2 ‖}, i. e., 𝐸󸀠 is an 𝐴𝑀-space. Let 𝑓 ∈ 𝐸󸀠+ now be such that ‖𝑓‖ = 1 and 𝑦1 , 𝑦2 ∈ 𝐹+ . Then 𝑓 ⊗ (𝑦1 ∨ 𝑦2 ) = (𝑓 ⊗ 𝑦1 ) ∨ (𝑓 ⊗ 𝑦2 ). The rank one operators 𝑇𝑖 = 𝑓 ⊗ 𝑦𝑖 , (𝑖 = 1, 2) belong to V(𝐸, 𝐹). This time we get ‖𝑦1 ∨ 𝑦2 ‖ = ‖𝑓 ⊗ (𝑦1 ∨ 𝑦2 )‖𝑟 = ‖(𝑓 ⊗ 𝑦1 ) ∨ (𝑓 ⊗ 𝑦2 )‖𝑟 = max{‖𝑇1 ‖𝑟 , ‖𝑇2 ‖𝑟 } = max{‖𝑦1 ‖‖𝑦2 ‖}, i. e., 𝐹 is an 𝐴𝑀-space. Observe that Theorem 4.47 does not contradict the fact that Orth(𝐸) is an 𝐴𝑀-space for arbitrary Banach lattices 𝐸. The reason is that Orth(𝐸), in general, does not contain all rank one operators. Theorem 4.48. Let 𝐸 and 𝐹 be Banach lattices and let V(𝐸, 𝐹) be a Banach lattice of operators with respect to the regular norm that contains all rank one operators. Then the following statements are equivalent: (1) V(𝐸, 𝐹) is an 𝐴𝑀-space with a rank one operator as an order unit; (2) 𝐸 is an 𝐴𝐿-space and 𝐹 is an 𝐴𝑀-space with order unit. Proof. (2) ⇒ (1) follows from Theorem 4.34 and Remark 4.39. For (1) ⇒ (2), it suffices to show that 𝐹 has an order unit since, by Theorem 4.47, 𝐸 is an 𝐴𝐿-space and 𝐹 an 𝐴𝑀-space. Denote the order unit of V(𝐸, 𝐹) by 𝑓⊗𝑢, where 𝑓 ∈ 𝐸󸀠+ and 𝑢 ∈ 𝐹+ . Then for each 𝑇 ∈ V(𝐸, 𝐹) there is some 𝛼 > 0 such that |𝑇| ≤ 𝛼(𝑓 ⊗ 𝑢). It is clear that 𝑓 ⊗ 𝑢 ≠ 0 since V(𝐸, 𝐹) contains all rank one operators. In particular 𝑓 ≠ 0. For arbitrary 𝑦 ∈ 𝐹, a number 𝛼𝑦 > 0 exists such that |𝑓 ⊗ 𝑦| ≤ 𝛼𝑦 (𝑓 ⊗ 𝑢). Then for 𝑥 ∈ 𝐸+ one has |𝑓 ⊗ 𝑦|(𝑥) = 𝑓(𝑥)|𝑦| ≤ 𝛼𝑦 𝑓(𝑥)𝑢. There exists 𝑥0 ∈ 𝐸+ such that 𝑓(𝑥0 ) > 0 and 𝑓(𝑥0 )|𝑦| ≤ 𝛼𝑦 𝑓(𝑥0 )𝑢. That is |𝑦| ≤ 𝛼𝑦 𝑢, which in turn shows that 𝑢 is an order unit in 𝐹. Now we will study the consequences for the Banach lattices 𝐸 and 𝐹 if it is known that Φ1 (V(𝐸, 𝐹)) = V(𝐸, 𝐹). In Theorem 4.10 (2.) this question was answered for the special case V(𝐸, 𝐹) = L 𝑟 (𝐸, 𝐹). As was mentioned in Remark 4.11, the Banach lattice 𝐹 may fail to have an order unit if 𝐸 is an 𝐴𝐿-space, 𝐹 is an 𝐴𝑀-space, and Φ1 (L 𝑟 (𝐸, 𝐹)) = L 𝑟 (𝐸, 𝐹). Next we prove some isomorphic results, where finite elements are not involved yet. Finite elements are pulled up in the subsequent corollaries. Theorem 4.49. Let 𝐸 and 𝐹 be Banach lattices, and let V(𝐸, 𝐹) be a Banach lattice of operators with respect to the regular norm that contains all rank one operators. If V(𝐸, 𝐹) is lattice isomorphic to an 𝐴𝑀-space, then 𝐸 is lattice isomorphic to an 𝐴𝐿-space, and 𝐹 is lattice isomorphic to an 𝐴𝑀-space.

98 | 4 Finite elements in vector lattices of linear operators Proof. Since V(𝐸, 𝐹) is lattice isomorphic to an 𝐴𝑀-space, according to [95, Theorem 2.1.12], 𝐶 > 0 exists such that for arbitrary positive disjoint operators 𝑇1 , . . . , 𝑇𝑛 ∈ V(𝐸, 𝐹) one has ‖𝑇1 + ⋅ ⋅ ⋅ + 𝑇𝑛 ‖𝑟 ≤ 𝐶 max{‖𝑇1 ‖𝑟 , . . . , ‖𝑇𝑛 ‖𝑟 }. In order to show that 𝐹 is lattice isomorphic to an 𝐴𝑀-space we use the constant 𝐶, and show that for arbitrary disjoint elements 𝑦1 , . . . , 𝑦𝑛 ∈ 𝐹+ one has ‖𝑦1 + ⋅ ⋅ ⋅ + 𝑦𝑛 ‖ ≤ 𝐶 max{‖𝑦1 ‖, . . . , ‖𝑦𝑛 ‖}. Let therefore 𝑓 ∈ 𝐸󸀠+ with ‖𝑓‖ = 1. Then ‖𝑓 ⊗ 𝑦‖𝑟 = ‖𝑦‖ for each 𝑦 ∈ 𝐹. The positive rank one operators 𝑇𝑖 defined by 𝑇𝑖 = 𝑓 ⊗ 𝑦𝑖 , (𝑖 = 1, . . . , 𝑛) belong to V(𝐸, 𝐹) and are pairwise disjoint. Indeed, for 𝑖 ≠ 𝑘 the relation 𝑦𝑖 ∧ 𝑦𝑘 = 0 implies 𝑇𝑖 ∧ 𝑇𝑘 = (𝑓 ⊗ 𝑦𝑖 ) ∧ (𝑓 ⊗ 𝑦𝑘 ) = 𝑓 ⊗ (𝑦𝑖 ∧ 𝑦𝑘 ) = 0. It follows ‖𝑦1 + ⋅ ⋅ ⋅ + 𝑦𝑛‖ = ‖𝑓 ⊗ (𝑦1 + ⋅ ⋅ ⋅ + 𝑦𝑛 )‖𝑟 = ‖𝑓 ⊗ 𝑦1 + ⋅ ⋅ ⋅ + 𝑓 ⊗ 𝑦𝑛 ‖𝑟 = ‖𝑇1 + ⋅ ⋅ ⋅ + 𝑇𝑛‖𝑟 ≤ 𝐶 max{‖𝑇1 ‖𝑟 , . . . , ‖𝑇𝑛 ‖𝑟 } = 𝐶 max{‖𝑦1 ‖, . . . , ‖𝑦𝑛 ‖}. In order to conclude that 𝐹 is isomorphic to an 𝐴𝑀-space, again refer to [95, Theorem 2.1.12]. In the same manner we show that 𝐸󸀠 is lattice isomorphic to an 𝐴𝑀-space, which is sufficient for 𝐸 to be lattice isomorphic to an 𝐴𝐿-space. Indeed, let 𝑓1 , . . . , 𝑓𝑛 ∈ 𝐸󸀠+ and 𝑦 ∈ 𝐹+ with ‖𝑦‖ = 1. Then the positive rank one operators 𝑓𝑖 ⊗ 𝑦, (𝑖 = 1, . . . 𝑛) belong to V(𝐸, 𝐹) and are pairwise disjoint. One has ‖𝑓1 + ⋅ ⋅ ⋅ + 𝑓𝑛 ‖ = ‖(𝑓1 + . . . + 𝑓𝑛 ) ⊗ 𝑦‖𝑟 = ‖𝑓1 ⊗ 𝑦 + ⋅ ⋅ ⋅ + 𝑓𝑛 ⊗ 𝑦‖𝑟 ≤ 𝐶 max{‖𝑓1 ⊗ 𝑦‖𝑟 , . . . , ‖𝑓𝑛 ⊗ 𝑦‖𝑟 } = 𝐶 max{‖𝑓1 ‖, . . . , ‖𝑓𝑛 ‖}. This theorem is a generalization of a result established by Wickstead ([136, Theorem 3.2]), which is the isomorphic version of his theorem cited at the beginning of the paragraph. Of course, Theorem 4.49 holds if Φ1 (V(𝐸, 𝐹)) = V(𝐸, 𝐹) is assumed. Indeed, by Theorem 3.19, the last condition implies that V(𝐸, 𝐹) is lattice isomorphic to an 𝐴𝑀-space (and the principal bands possess generalized order units). As a special case we now get the result of the cited Theorem 4.10 (2). Corollary 4.50. Let 𝐸, 𝐹 be Banach lattices with 𝐹 Dedekind complete. If Φ1 (L 𝑟 (𝐸, 𝐹)) = L 𝑟 (𝐸, 𝐹), then for 𝐸 and 𝐹 the conclusions of the previous theorem hold. Corollary 4.51. Let 𝐸 be an atomic Banach lattice with order continuous norm and let 𝐹 be a Banach lattice. If Φ1 (L 𝑟 (𝐸, 𝐹)) = L 𝑟 (𝐸, 𝐹), then for 𝐸 and 𝐹 the conclusions of the previous theorem hold. Indeed, under the assumptions the space L 𝑟 (𝐸, 𝐹) is a Banach lattice by Theorem 4.30 (b). It is clear that all rank one operators belong to L 𝑟 (𝐸, 𝐹).

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Corollary 4.52. Let 𝐸 be a Banach lattice and 𝐹 an 𝐴𝑀-space. If Φ1 (K(𝐸, 𝐹)) = K(𝐸, 𝐹), then 𝐸 is isomorphic to an 𝐴𝐿-space. Indeed, under the assumptions the space K(𝐸, 𝐹) is a Banach lattice by Theorem 4.30 (f) and, of course, contains all rank one operators.

5 The space of maximal ideals of a vector lattice The concept of ideals defined in a set with a fixed structure, e. g., vector lattice, 𝑙-group, ℓ-algebra, ring, and others, has been proved to be very useful in investigating the given structure more deeply and to provide a useful tool for establishing some topological space which is necessary for the representation by means of continuous functions; see [25, 59, 84, 95, 97, 120]. In an Archimedean vector lattice 𝐸 consider the ideals (see Section 2.2). Clearly, {0} and 𝐸 are always ideals, the so-called trivial ideals. If the set of all maximal ideals (see Definition 5.3 in Section 5.2) is equipped with a suitable topology, then the corresponding topological space can be used to represent the vector lattice 𝐸, as a vector lattice of continuous functions on this topological space. In the vector lattice 𝐶(𝑄) of all real continuous functions on a topological space 𝑄, the set {𝑥 ∈ 𝐶(𝑄) : 𝑥(𝑠) = 0} is a maximal ideal for any fixed 𝑠 ∈ 𝑄. Moreover, if 𝑄 is compact and Hausdorff, it can even be shown that each maximal ideal is of such a kind (see [84, Example 27.7]).

5.1 Representation of vector lattices by means of extended real continuous functions Much important information on a vector lattice can be obtained if the latter is represented as a vector lattice of real-valued continuous functions on some topological space. We have already been faced with the representation problem for normed vector lattices in Section 2.5. As mentioned before, the proofs of the new results provided for some classes of Banach lattices essentially make use of the norm. If there is no norm in the vector lattice, but the latter is Dedekind complete, then representations by means of extended real-valued continuous functions on some compact topological space are possible and will be briefly reported next. The results provided in this section demonstrate that for all Archimedean vector lattices, such lattice isomorphic representations on appropriate compact topological spaces are possible. We will briefly rest on the basic constructions and provide the related main theorems. Let 𝐸 be an Archimedean vector lattice and 𝑄 a topological Hausdorff space. A vector lattice 𝐸(𝑄) consisting of continuous (not necessarily everywhere finite) functions on 𝑄 is termed a representation of the vector lattice 𝐸, if there is a Riesz isomorphism 𝑖 : 𝐸 → 𝐸(𝑄). More exactly, a representation of 𝐸 should be understood as a pair (𝐸(𝑄), 𝑖), where 𝐸(𝑄) is a vector lattice of continuous functions on some compact Hausdorff space 𝑄, and 𝑖 : 𝐸 → 𝐸(𝑄) is a Riesz isomorphism. Notice that later (from

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Chap. 7 on) we deal with representations of vector lattices by means of continuous functions with everywhere finite (real) values; see Definition 7.1. Let ℝ∞ = ℝ∪{−∞, +∞} be the extended real line comprised of the usual extended algebraic operations and ordered by −∞ < 𝑥 < ∞ for 𝑥 ∈ ℝ. For a topological space 𝑄 denote by 𝐶∞ (𝑄) the set of all extended real-valued continuous functions 𝑥 : 𝑄 → ℝ∞ , i. e., continuous functions on 𝑄, each of which may attain the values +∞ and −∞ on a nowhere dense subset of 𝑄. A compact topological Hausdorff space¹ 𝑄 is called extremally disconnected² if the closure of any open set in 𝑄 is an open, i. e., a closed-open, set. This property of 𝑄 is necessary and sufficient for the vector lattice 𝐶(𝑄) to be Dedekind complete; see e. g., [9, 109, 120]. If 𝑄 is an extremally disconnected compact space, then the set 𝐶∞ (𝑄) can be linearized and equipped with its natural order, where 𝑥 ≤ 𝑦 if 𝑥(𝑡) ≤ 𝑦(𝑡) for any 𝑡 ∈ 𝑄. Then the space 𝐶∞ (𝑄) becomes a Dedekind complete vector lattice; see [59, 120]. An ideal 𝐼 in a vector lattice 𝐸 is called an order dense ideal or a fundament³ if 𝐼 is a complete set in 𝐸, i. e., 𝑥 ⊥ 𝐼 implies 𝑥 = 0 (see [144, § 23]). For Dedekind complete vector lattices, the following representation result is very important (see [120, Theorem V.4.2]). Theorem 5.1. For each Dedekind complete vector lattice 𝐸 an extremally disconnected compact space 𝑄 exists such that 𝐸 is Riesz isomorphic to a fundament 𝐸∞ (𝑄) in 𝐶∞(𝑄). Moreover, the space 𝑄 is defined uniquely up to a homeomorphism and, if 𝐸0 is a given complete collection of pairwise disjoint positive elements in 𝐸, then the isomorphism can be constructed such that the elements of 𝐸0 are represented as the characteristic functions of the closed-open subsets in 𝑄. The complete proof of this theorem can be found in [120]. We will mention here only the main steps, which will give us the possibility to analyse in more detail some aspects of the representations and will prepare the background for later considerations. First of all, Zorn’s Lemma guarantees that in each vector lattice a complete set of pairwise disjoint positive elements exists (see [120, Lemma IV.7.1]). This leads to the fact that each Dedekind complete vector lattice 𝐸 possesses a decomposition into a complete system of pairwise disjoint bands {𝐸𝜉 }𝜉∈Ξ , each of which itself is a Dedekind complete vector lattice with an order unit, say 1𝜉 , and such that for any 𝑥 ∈ 𝐸 there exists its projection onto each band 𝐸𝜉 . By means of the two systems {𝐸𝜉 }𝜉∈Ξ and (1𝜉 )𝜉∈Ξ , a Dedekind complete vector lattice 𝑌 with an order unit 1 can be constructed such that 𝐸 is lattice isomorphic to some fundament in 𝑌. For the Dedekind complete vector lattice 𝑌 with order unit 1, the sublattice E(𝑌) of 𝑌 consisting of all elements 𝑒 ∈ 𝑌,

1 Usually the Hausdorff axiom is included in the definition of a compact space. 2 Or Stonean in [109] and [2]. 3 In [120] and [84] also known as a foundation.

102 | 5 The space of maximal ideals of a vector lattice such that 𝑒 ∧ (1 − 𝑒) = 0 (components of 1), is a complete⁴ Boolean algebra. In virtue of M. Stone’s Theorem (see [77, Theorem 1.2.3], [120, Theorem II.9.1 9], [9, Theorem 12.25]), a uniquely (up to homeomorphism) defined, extremally disconnected compact Hausdorff space⁵ 𝑄 exists, whose collection of open-closed subsets is isomorphic to E(𝑌). Then 𝑌 is lattice isomorphic to some fundament 𝑌󸀠 in 𝐶∞(𝑄). Since 𝐸 is a fundament in 𝑌, it can be considered also to be a fundament 𝐸∞ (𝑄) in 𝐶∞ (𝑄). The isomorphism can be chosen such that the image of 1 ∈ 𝑌 is the function equal to 1 for any 𝑡 ∈ 𝑄, and that 𝐶(𝑄) ⊂ 𝑌󸀠 . As mentioned in Section 2.2, any Archimedean vector lattice possesses its Dedekind completion. More exactly, if 𝐸 is an Archimedean vector lattice, then (up to a lattice isomorphism) a uniquely defined Dedekind complete vector lattice 𝐸𝛿 exists with the properties recorded in Section 2.2. This fact allows us to apply Theorem 5.1 to 𝐸𝛿 and to get a representation for the Archimedean vector lattice 𝐸 as well. We provide this result in the convenient formulation given in [2, Theorem 3.35] (for details, the reader is referred to [86], [120, Theorem V.7.1], [84, Chap. 7], and [134]). Theorem 5.2 (F. Maeda, T. Ogasawara, B. Z. Vulikh). For each Archimedean vector lattice 𝐸 a unique (up to homeomorphism) extremally disconnected compact topological Hausdorff space 𝑄 and an order dense vector sublattice 𝐸∞ (𝑄) ⊂ 𝐶∞ (𝑄) exist such that 𝐸 and 𝐸∞ (𝑄) are lattice isomorphic. The isomorphism preserves all suprema and infima. Moreover, (1) if 𝐸 has a weak order unit 𝑒, then the lattice isomorphism can be chosen to carry the element 𝑒 to the constant function 1 on 𝑄; (2) 𝐸∞ (𝑄) is an (order dense) ideal in 𝐶∞(𝑄) if and only if 𝐸 is Dedekind complete; (3) if 𝐸 is Dedekind complete and has a (strong) order unit 1, then 𝐸∞ (𝑄) = 𝐶(𝑄). The extremally disconnected compact space 𝑄 is called the canonical compact or the Stone space for 𝐸. Following [2], the uniquely constructed Dedekind complete vector lattice 𝐶∞(𝑄) is termed the Maeda-Ogasawara-Vulikh completion , or universal completion of 𝐸. On 𝑄 different representations of 𝐸 are possible. Each of them is determined by the choice of the complete system of pairwise disjoint positive elements in 𝐸. Namely, the order unit 1 constructed above in the space 𝑌 depends on that system. Then, in turn, the unit function on 𝑄 is assigned to this order unit 1. For further purposes we need only a subspace S(𝐸) ⊆ 𝑄, the so-called proper space⁶ of 𝐸 (see [116, 117]), which is defined as the set S(𝐸) of those 𝑡 ∈ 𝑄 for which there exists 𝑥 ∈ 𝐸 with 𝑥(𝑡) ≠ 0 for at least one representation of 𝐸 on 𝑄. For an element 𝑥 ∈ 𝐸, the set 𝑄𝑥 = {𝑡 ∈ 𝑄 : 𝑥(𝑡) ≠ 0} is both open and closed (and compact). It

4 I. e., every nonempty subset has supremum and infimum. 5 The actual points of the set 𝑄 are the maximal ideals of Boolean algebra E(𝑌). 6 Or Nakano space if 𝐸 is Dedekind complete; see [100].

5.2 Maximal ideals and discrete functionals

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does not depend on the chosen representation, and it is clear that 𝑄𝑥 ⊂ S(𝐸) for each 𝑥 ∈ 𝐸. It is easy to see that for any 𝑡0 ∈ S(𝐸) there is an element 0 ≠ 𝑥0 ∈ 𝐸+ , such that 𝑥0 (𝑡0 ) > 0 holds for at least one representation of 𝐸 on 𝑄. Following [117] we write for this obstacle 𝑥0 [𝑡0 ] > 0. For a point 𝑡0 ∈ 𝑄 and an element 𝑥 ∈ 𝐸 we write 𝑥[𝑡0 ] = 0, if 𝑥(𝑡 ) for any representation of 𝐸 it holds 𝑥(𝑡0 ) = 0. For given 𝑥 ∈ 𝐸, the relation 𝑥 (𝑡0 ) has one 0 0 and the same value for any representation of 𝐸, whenever 𝑥0 (𝑡0 ) ≠ 0, ±∞. This value as an element of ℝ∞ will be denoted by 𝑥𝑥 [𝑡0 ]. 0 Observe that representations of vector lattices constructed by means of Stone’s Theorem are targeted on representing the vector lattice by means of continuous functions on compact topological spaces. The price which is paid is the appearance of extended real-valued continuous functions. In Chapter 7 and later we will consider representations of vector lattices by means of continuous functions with finite values everywhere. Then the price for that approach will be the loss of compactness (at least the local compactness is preserved), which, however, seems to be acceptable if the finite elements can be represented as functions with compact support.

5.2 Maximal ideals and discrete functionals We start with the definition of a maximal ideal in a vector lattice. Definition 5.3. A nontrivial ideal in a vector lattice 𝐸 is said to be maximal if it is not contained in any other ideal different from 𝐸. If one is guided by certain analogy with properties of ideals in algebras or rings, then the natural and important problems for applications are the existence and the characterization of maximal ideals as well as their sufficiency, i. e., when every ideal of 𝐸 is contained in a maximal ideal of 𝐸. In an arbitrary Archimedean vector lattice 𝐸, the trivial ideals {0} and 𝐸 may be the only ideals at all. In this case, the vector lattice 𝐸 is Riesz isomorphic to ℝ and is called simple. In general, not every ideal is contained in a maximal ideal. Moreover, in 𝐸 there might not be maximal ideals at all; see [84, Example 27.8], [117], and [64]. However, if 𝐸 has an order unit 1 , then each ideal can be extended to a maximal ideal. Indeed, if the collection I(𝐸) of all proper ideals in 𝐸 is ordered by inclusion, then no ideal of I(𝐸) contains 1. Any chain of ideals in I(𝐸) is bounded in I(𝐸), since the union J of all ideals of the chain is an ideal in 𝐸 which is proper, due to 1 ∉ J. According to Zorn’s Lemma, each element of I(𝐸) is followed by a maximal one. Therefore, any proper ideal of 𝐸 lies in some maximal one. The maximal ideals in an Archimedean vector lattice 𝐸 can be characterized by means of strong points of the proper space S(𝐸). A point 𝑡0 ∈ S(𝐸) is said to be strong, if a positive element 0 < 𝑥0 ∈ 𝐸 exists, such that 𝑥0 (𝑡0) > 0 for some representation of 𝐸, and 𝑥𝑥 [𝑡0 ] ≠ ±∞ for any 𝑥 ∈ 𝐸. Such an element 𝑥0 is called a strong local unit at 𝑡0 . 0

104 | 5 The space of maximal ideals of a vector lattice Now we are able to formulate two basic results of A. I. Veksler on the general form of a maximal ideal in an Archimedean vector lattice, and on the characterization of vector lattices with a sufficient number of maximal ideals. Theorem 5.4 ([117, Theorem 1]). Let 𝐸 be an Archimedean vector lattice and 𝑡0 a strong point of S(𝐸). Then the set 𝑀𝑡0 = {𝑥 ∈ 𝐸 :

𝑥 [𝑡 ] = 0}, 𝑥0 0

where 𝑥0 is a strong local unit at 𝑡0 , is a maximal ideal in 𝐸 and does not depend on the choice of 𝑥0 . Vice versa, each maximal ideal in 𝐸 has such structure. Further on, in this and in the next chapters, we will consider vector lattices in which each proper ideal is contained in some maximal ideal. Such vector lattices are called vector lattices with sufficiently many maximal ideals in [116, 117]. Theorem 5.5 ([117, Theorem 2]). In an Archimedean vector lattice 𝐸, each proper ideal is contained in some maximal ideal if and only if its proper space S(𝐸) consists only of strong points and, consequently, a strong local unit exists for each point of S(𝐸) . It is clear that a vector lattice of bounded elements, i. e., a vector lattice with a strong order unit, satisfies the condition of the theorem. The Kaplansky vector lattice is a vector lattice without a strong order unit which possesses a sufficient number of maximal ideals (see [117, § 6]). For a given Archimedean vector lattice 𝐸 denote by M(𝐸), sometimes abbreviated to M, the set of all maximal ideals of the vector lattice 𝐸. We introduce now a class of linear functionals on vector lattices, the kernels of which are maximal ideals. Definition 5.6. A nonzero linear functional 𝑓 : 𝐸 → ℝ on the vector lattice 𝐸 is called discrete, if for any 𝑥, 𝑦 ∈ 𝐸, one has⁷ 𝑓(𝑥 ∨ 𝑦) = max{𝑓(𝑥), 𝑓(𝑦)} and

𝑓(𝑥 ∧ 𝑦) = min{𝑓(𝑥), 𝑓(𝑦)}.

(5.1)

Any (nonzero) discrete functional can be considered as a Riesz homomorphism with values in ℝ, indeed is a discrete element in 𝐸̃ (see e. g., [139, Proposition 0.3.8]), and therefore is always positive and (𝑟)-continuous, i. e., continuous with respect to the (𝑟)-convergence (see p. 10). Since in any vector lattice 𝐸, for all 𝑥, 𝑦 ∈ 𝐸, there holds the equation 𝑥 + 𝑦 = 𝑥 ∨ 𝑦 + 𝑥 ∧ 𝑦, (5.2) a nonzero linear functional 𝑓 on 𝐸 is already discrete if at least one of the equations (5.1) is satisfied.

7 These equations are also written as 𝑓(𝑥 ∨ 𝑦) = 𝑓(𝑥) ∨ 𝑓(𝑦) and 𝑓(𝑥 ∧ 𝑦) = 𝑓(𝑥) ∧ 𝑓(𝑦).

5.2 Maximal ideals and discrete functionals

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105

Denote the set of all discrete functionals of the vector lattice 𝐸 by Δ(𝐸). For any discrete functional 𝑓 ≠ 0, its kernel 𝑓−1 (0) is in M(𝐸) and, vice versa, any 𝑀 ∈ M(𝐸) defines a discrete functional (up to a constant coefficient). Theorem 5.7 ([77, Theorem 3.3.1(3)]). The maximal ideals in a vector lattice are the kernels of the real valued Riesz homomorphisms on 𝐸. If 𝐸(𝑄) is a vector lattice of continuous functions on a topological Hausdorff space 𝑄 which satisfies the condition⁸ (⋆)

for any point 𝑡 ∈ 𝑄 there is a function 𝑥 ∈ 𝐸(𝑄) with 𝑥(𝑡) ≠ 0,

then the simplest discrete functionals on 𝐸(𝑄) are 𝛿𝑡 (𝑥) = 𝑥(𝑡), 𝑥 ∈ 𝐸(𝑄), where 𝑡 is an arbitrary fixed point of 𝑄. The set K([0, ∞)) is an ideal in the vector lattice 𝐶lim ([0, ∞)) of all continuous functions on [0, ∞), possessing a finite limit for 𝑡 → ∞. However, this ideal is not maximal since K([0, ∞)) ⫋ 𝑓−1 (0) for the discrete functional 𝑓, defined as 𝑓(𝑥) = lim 𝑥(𝑡) for each 𝑥 ∈ 𝐶lim ([0, ∞)). The corresponding maximal in which K([0, ∞)) is 𝑡→∞

included is the ideal 𝐶0 ([0, ∞)) of all continuous functions 𝑥 with lim𝑡→∞ 𝑥(𝑡) = 0. As mentioned above, in an arbitrary vector lattice maximal ideals, and therefore also nontrivial discrete functionals, might not exist at all. According to Theorem 5.4, their existence is related to the availability of strong points in S(𝐸); see also [64]. It is clear that to the maximal ideal 𝑀𝑡0 , mentioned in Theorem 5.4, there are assigned all discrete functionals 𝜆𝑓𝑡0 ,𝑥0 for 𝜆 ≠ 0, where 𝑓𝑡0 ,𝑥0 (𝑥) := 𝑥𝑥 [𝑡0 ]. 0 Concerning maximal ideals or discrete functionals in a vector lattice 𝐸, we assume not only the existence of sufficient numbers of them, but also that there are enough in order to separate the elements, i. e., for any 0 ≠ 𝑥 ∈ 𝐸 a maximal ideal 𝑀 exists such that 𝑥 ∉ 𝑀, or equivalently, that 𝐸 is radical-free or semi-simple. The latter means that 𝐸 has a trivial radical, i. e., the 𝑅 = 𝑅(𝐸) = ⋂{𝑀 : 𝑀 ∈ M(𝐸)} = {0}. A vector lattice is radical-free if and only if the set Δ(𝐸) of all discrete functionals is total on 𝐸, i. e., if 𝑓(𝑥) = 0 for all 𝑓 ∈ Δ(𝐸) then 𝑥 = 0. It is well-known that an Archimedean Riesz space with an order unit is radical-free (see [77, 1.3.7.(3)], [84, Theorem 27.6], and also the proof of [120, Theorem VII.5.1]). The very important class of vector lattices of type (Σ) is radical-free as well. This follows from the theorem below. Theorem 5.8. In each vector lattice 𝐸 of type (Σ) the set Δ(𝐸) is total on 𝐸. Proof. Let (𝑒𝑛)𝑛∈ℕ be a sequence in 𝐸 which satisfies the condition (Σ󸀠 ). According to Theorem 5.2, a compact topological space 𝑄 exists such that 𝐸 is order isomorphic to some vector lattice 𝐸∞ (𝑄) of extended continuous functions on 𝑄. Denote by 𝑄𝑛 the nowhere dense subset of 𝑄, where the function 𝑒𝑛(𝑡), the image of 𝑒𝑛 under the

8 See also p. 144.

106 | 5 The space of maximal ideals of a vector lattice ∞

isomorphism between 𝐸 and 𝐸∞ (𝑄), takes on infinite values. The set ⋃𝑛=1 𝑄𝑛 is of first category. This implies that the set of discrete functionals of the kind 𝛿𝑡 with 𝛿𝑡 (𝑥) = 𝑥(𝑡) for 𝑡 ∈ 𝑄 \ ⋃∞ 𝑛=1 𝑄𝑛 is total on 𝐸. An equivalent formulation of the theorem is Corollary 5.9. Each vector lattice of type (Σ) is radical-free. In the Archimedean vector lattice consisting of all functions on the interval [0, 1] which are continuous everywhere except at a finite number of points, where they can have a pole of second order, the intersection of the maximal ideals consists of all continuous functions. This vector lattice was first considered by I. Kaplansky (see [25, § XV.3]), and provided the reason for constructing the vector lattices⁹ of the type (Σ) (and therefore radical-free vector lattices) in our Examples 3.4 and 3.5. If 𝐸 is vector lattice with 𝑅 = 𝑅(𝐸) ≠ {0}, the vector lattice 𝐸/𝑅 is already radicalfree and will be considered instead of 𝐸. Proposition 5.10. Let be 𝐸 a vector lattice, 0 ≤ 𝑥, 𝑦 ∈ 𝐸 and 𝑓 ∈ Δ(𝐸). Then 𝑓(𝑥)𝑓(𝑦) ≠ 0 implies 𝑓(𝑥 ∧ 𝑦) ≠ 0. Proof. If 𝑓(𝑥 ∧ 𝑦) = 0, then by (5.2) one would have 𝑓(𝑥) + 𝑓(𝑦) = max{𝑓(𝑥), 𝑓(𝑦)}, which is impossible due to the assumption that both numbers 𝑓(𝑥), 𝑓(𝑦) are positive. It is clear that under the assumptions of the proposition one has 𝑥 ∧ 𝑦 ≠ 0. Moreover, 𝑓(𝑥 ∧ 𝑦) = 0 implies that at least one of the numbers 𝑓(𝑥), 𝑓(𝑦) must be zero. This shows that each maximal ideal 𝑀 in 𝐸 is also a prime ideal, i. e., 𝑥, 𝑦 ∈ 𝐸, and 𝑥∧𝑦 = 0 implies that at least one of 𝑥 ∈ 𝑀 or 𝑦 ∈ 𝑀 holds; see[84, § 33]. The inverse implication is not true, i. e., not every prime ideal is maximal; see Example 5.19. For maximal ideals the previous proposition is equivalent to the following. Corollary 5.11. If the vectors 𝑥, 𝑦 ∈ 𝐸 do not belong to some maximal ideal 𝑀, then {𝑥󸀠 ∈ 𝐸 : 𝑥󸀠 ≥ 𝑥 ∧ 𝑦} ∩ 𝑀 = 0. A subset 𝐴 in a vector lattice 𝐸 is called sufficient if for each functional 𝑓 ∈ Δ(𝐸), there is an element 𝑎 ∈ 𝐴 such that 𝑓(𝑎) ≠ 0. It is easy to see that a set 𝐴 is sufficient if and only if for an arbitrary maximal ideal 𝑀 ∈ M(𝐸) there is an element 𝑎 ∈ 𝐴 with 𝑎 ∉ 𝑀. Corollary 5.12. Let 𝐸 be a vector lattice with a total set Δ(𝐸). Then any sufficient subset of 𝐸 is also complete. Indeed, if an element 𝑧 is disjoint to a sufficient set 𝐴, then for 𝑓 ∈ Δ(𝐸) there is an element 𝑎 ∈ 𝐴 with 𝑓(|𝑎|) > 0. From 𝑎 ⊥ 𝑧 one has 𝑓(|𝑎|) ∧ 𝑓(|𝑧|) = 𝑓(|𝑎| ∧ |𝑧|) = 0, which implies 𝑓(|𝑧|) = 0. Since Δ(𝐸) is total, 𝑧 = 0 follows.

9 Named Kaplansky vector lattices.

5.3 The topology on the space of maximal ideals of a vector lattice

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5.3 The topology on the space of maximal ideals of a vector lattice The collection of all maximal ideals in rings and algebras is usually equipped with a special topology which provides an additional tool in order to describe or characterize properties of the ring or algebra; see [80]. For any Archimedean vector lattice we introduce now a topological Hausdorff space and assign to each vector some closed subset in this space. Then it will be established that exactly the finite elements generate compact subsets in this space, i. e., the compactness of those sets characterizes the finite elements. This space also carries much information about the vector lattice, and provides an appropriate ambience for further investigations and also for representations of vector lattices, such that each finite element is represented as a continuous function with compact support. The collection M(𝐸) of all maximal ideals of a vector lattice 𝐸 with sufficient maximal ideals will be equipped with a suitable topology by means of defining the closure for any nonempty subset M ⊂ M(𝐸). The set 𝑘(M) := ⋂𝑀∈M 𝑀 is the kernel of the set M. Denote by ℎ(𝑘(M)) = {𝑀 ∈ M(𝐸) : 𝑀 ⊃ 𝑘(M)} the hull of the set 𝑘(M), i. e., the set of all maximal ideals each which contains 𝑘(M). Then by M := ℎ(𝑘(M)) a certain operation on the collection of all subsets of M(𝐸) is defined, which satisfies the axioms of closed sets (see [75]), and therefore defines a topology, the so-called hull-kernel topology 𝜏ℎ𝑘 on M(𝐸) (see [80, § 4]). So, for any vector lattice 𝐸 with a sufficient number of maximal ideals we have now introduced the topological space (M(𝐸), 𝜏ℎ𝑘 ) of all maximal ideals. In particular, we will often make use of the following: a maximal ideal 𝑀0 ∈ M(𝐸) belongs to the closure of a set M, i. e., 𝑀0 ∈ M (or, equivalently, 𝑀0 is a closure point of the set M), if and only if 𝑀0 ⊃ ⋂ 𝑀. 𝑀∈M

Important subsets of M(𝐸) are obtained as follows. For any 𝑥 ∈ 𝐸 put 𝐺𝑥 = {𝑀 ∈ M(𝐸) : 𝑥 ∉ 𝑀}. Then the set suppM (𝑥) := 𝐺𝑥 is called the abstract support of the element 𝑥. Observe that 𝐺𝑥 = 𝐺|𝑥| for any 𝑥 ∈ 𝐸. This follows from 𝐼𝑥 = 𝐼|𝑥| and the relation 𝐺𝑥 = 𝐺(𝐼𝑥 ) = ⋃ {𝐺𝑦 : 𝑦 ∈ 𝐼𝑥 } mentioned in [90], where for any nonempty 𝐴 ⊂ 𝐸 we put 𝐺(𝐴) = ⋃ 𝐺𝑥 . 𝑥∈𝐴

For 𝐴 ⊂ 𝐸 put 𝐺(𝐴) = ⋃𝑥∈𝐴 𝐺𝑥 . Proposition 5.13. Let 𝐸 be a vector lattice and 0 ≤ 𝑥, 𝑦 ∈ 𝐸. For 𝑥 ⊥ 𝑦 the condition 𝐺𝑥 ∩ 𝐺𝑦 = 0 is necessary, and in case of a radical-free 𝐸 also sufficient.

108 | 5 The space of maximal ideals of a vector lattice Proof. If 𝐺𝑥 ∩ 𝐺𝑦 ≠ 0, then a maximal ideal 𝑀 exists such that 𝑥 ∉ 𝑀, 𝑦 ∉ 𝑀, and according to Corollary 5.11 one has 𝑥 ∧ 𝑦 ∉ 𝑀. This is a contradiction, since 𝑥 ∧ 𝑦 = 0 and 0 belongs to 𝑀 for each maximal ideal. If 𝑅(𝐸) = {0} and 𝐺𝑥 ∩ 𝐺𝑦 = 0, then by assuming 𝑥 ∧ 𝑦 > 0, there is a maximal ideal 𝑀 such that 𝑥 ∧ 𝑦 ∉ 𝑀. However, since 𝑀 is an ideal in 𝐸, this implies 𝑥 ∉ 𝑀, 𝑦 ∉ 𝑀. So 𝑀 ∈ 𝐺𝑥 ∩ 𝐺𝑦 . The main properties of the topology 𝜏ℎ𝑘 are collected in the following theorem. Theorem 5.14. Let 𝐸 be a vector lattice and M = (M(𝐸), 𝜏ℎ𝑘 ) its space of maximal ideals¹⁰. Then (1) for each 𝑥 ∈ 𝐸 the set 𝐺𝑥 is open with respect to 𝜏ℎ𝑘 ; (2) the system {𝐺𝑥}𝑥∈𝐸 is the basis for the topology 𝜏ℎ𝑘 in M; (3) let 𝐺 be an open subset, and 𝐾 a compact subset of M such that 𝐾 ⊂ 𝐺. Then an element 0 ≤ 𝑧 ∈ 𝐸 exists such that 𝐾 ⊂ 𝐺𝑧 ⊂ 𝐺; (4) let 𝑓𝛼 , 𝑓 ∈ Δ(𝐸), where 𝑓𝛼 → 𝑓 with respect to the weak topology 𝜎(𝐸,̃ 𝐸). Then 𝑓𝛼−1 (0) → 𝑓−1 (0) with respect to the toplogy 𝜏ℎ𝑘 ; (5) a subset M ⊂ M is dense in the space M if and only if ⋂{𝑀 : 𝑀 ∈ M} = ⋂{𝑀 : 𝑀 ∈ M}. Proof. (1) We prove that the set 𝐹𝑥 = M \ 𝐺𝑥 is closed. Let 𝑀󸀠 be a closure point of the set 𝐹𝑥 , i. e., 𝑀󸀠 ⊃ ⋂{𝑀 : 𝑀 ∈ 𝐹𝑥 }. Then 𝑥 ∈ 𝑀󸀠 and so 𝑀󸀠 ∈ 𝐹𝑥 . (2) Let 𝑀󸀠 be a point of M and 𝑈 an open neighborhood of 𝑀󸀠 . Since 𝑈 is open, 󸀠 𝑀 cannot be a closure point for the set M \ 𝑈, and therefore 𝑀󸀠 ⊅ ⋂{𝑀 : 𝑀 ∉ 𝑈}. Consequently, for some 𝑦 ∈ 𝐸 there hold the relations 𝑦 ∈ ⋂{𝑀 : 𝑀 ∉ 𝑈} and

𝑦 ∉ 𝑀󸀠 ,

which means 𝑀󸀠 ∈ 𝐺𝑦 ⊂ 𝑈. (3) Any point 𝑀 ∈ 𝐾 is an interior point of 𝐺. According to statement (2) there is an element 𝑥𝑀 ∈ 𝐸 with 𝑥𝑀 > 0 and 𝑀 ∈ 𝐺𝑥𝑀 ⊆ 𝐺. The sets 𝐺𝑥𝑀 for 𝑀 ∈ 𝐾 form an open covering of 𝐾. Due to the compactness of 𝐾 for a finite number of elements 𝑥𝑀1 , 𝑥𝑀2 , . . . , 𝑥𝑀𝑛 , the inclusion 𝐾 ⊆ ⋃𝑛𝑖=1 𝐺𝑥𝑀 ⊆ 𝐺 holds. For the element 𝑧 = 𝑥𝑀1 ∨ 𝑖 𝑥𝑀2 ∨ ⋅ ⋅ ⋅ ∨ 𝑥𝑀𝑛 we then have 𝐺𝑧 = ⋃𝑛𝑖=1 𝐺𝑥𝑀 . 𝑖

(4) Let 𝑀𝛼 = 𝑓𝛼−1 (0), 𝑀 = 𝑓−1 (0), and 𝑈 an open neighborhood of 𝑀. According to statement (2) there is an element 𝑧 ∈ 𝐸 with 𝑀 ∈ 𝐺𝑧 ⊆ 𝑈. The element 𝑧 does not belong to 𝑀, which is equivalent to 𝑓(𝑧) ≠ 0. By assumption there is an index 𝛼0 , such that 𝑓𝛼 (𝑧) ≠ 0 for all 𝛼 ≥ 𝛼0 . This means 𝑧 ∉ 𝑀𝛼 for 𝛼 ≥ 𝛼0 . Consequently, 𝑀𝛼 ∈ 𝐺𝑧 ⊆ 𝑈 whenever 𝛼 ≥ 𝛼0 .

10 In the present and following chapter, discussing the space M, we assume that the underlying vector lattice has a sufficient number of maximal ideals, as mentioned above.

5.4 The Hausdorff property of M |

109

(5) Let the subset M be dense in the M. Then any point 𝑀0 ∈ M is a closure point of the set M, i. e., 𝑀0 ⊇ ⋂{𝑀 : 𝑀 ∈ M}. Therefore one has ⋂{𝑀 : 𝑀 ∈ M} ⊇ ⋂{𝑀 : 𝑀 ∈ M}. The inverse inclusion is obvious, which shows that ⋂{𝑀 : 𝑀 ∈ M} = ⋂{𝑀 : 𝑀 ∈ M} holds. If for some subset M ⊂ M the last equality is satisfied, then the set ⋂{𝑀 : 𝑀 ∈ M} is contained in any maximal ideal. Therefore, each maximal ideal is a closure point of the set M. Corollary 5.15. (1) Let M ⊆ M(𝐸) and 𝑀0 ∈ int(M).. Then there is an element 0 ≤ 𝑧 ∈ 𝐸 such that 𝑀0 ∈ 𝐺𝑧 ⊆ M. (2) If the vector lattice 𝐸 is radical-free then a subset M ⊂ M is dense in M if and only if ⋂{𝑀 : 𝑀 ∈ M} = {0}. (3) In a radical-free vector lattice 𝐸 a subset 𝐴 is complete¹¹ in 𝐸 if and only if the set 𝐺(𝐴) is dense in M Proof. (1) and (2) follow immediately from statements (2) and (5) of the theorem. (3) If a set 𝐴 ⊂ 𝐸 is complete in 𝐸 then, by assuming 𝐺(𝐴) ≠ M, there is maximal ideal 𝑀0 ∈ M such that 𝑀0 ⊅ ⋂{𝑀 : 𝑀 ∈ 𝐺(𝐴)}. It then follows that there is an element 𝑦 ≠ 0 with 𝑦 ∈ ⋂{𝑀 : 𝑀 ∈ 𝐺(𝐴)}. We show now that the vector 𝑦 is disjoint to each element of 𝐴. For that, due to the assumption 𝑅(𝐸) = {0}, it suffices to check 󵄨 󵄨 󵄨 󵄨 whether 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ∧ |𝑥| ∈ 𝑀 for each 𝑀 ∈ M and 𝑥 ∈ 𝐴. If 𝑀 ∈ 𝐺(𝐴), then 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ∧ |𝑥| ∈ 𝑀 󵄨 󵄨 󵄨 󵄨 because of 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ∈ 𝑀. If 𝑀 ∉ 𝐺(𝐴), then 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ∧ |𝑥| ∈ 𝑀 because of |𝑥| ∈ 𝑀. So we obtained 𝑦 ⊥ 𝐴, which contradicts 𝑦 ≠ 0, since 𝐴 is complete. Let now 𝐺(𝐴) be dense in M and 𝑦 ⊥ 𝐴. If 𝑀 ∈ 𝐺(𝐴), then there is some vector 󵄨 󵄨 𝑥 ∈ 𝐴 with 𝑥 ∉ 𝑀. Together with 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ∧ |𝑥| = 0 ∈ 𝑀 it follows 𝑦 ∈ 𝑀. So we get 𝑦 ∈ 𝑀 for each maximal ideal 𝑀 ∈ 𝐺(𝐴), i. e., 𝑦 ∈ ⋂{𝑀 : 𝑀 ∈ 𝐺(𝐴)}. On the other hand, by (2) of the corollary one has ⋂{𝑀 : 𝑀 ∈ 𝐺(𝐴)} = {0}. This shows that 𝑦 = 0. Later we demonstrate how the knowledge of M(𝐸) can be used for the determination of the finite and selfmajorizing elements in a vector lattice 𝐸 (see Example 6.28).

5.4 The Hausdorff property of M Further on, 𝐸 is assumed to be a radical-free vector lattice. Theorem 5.16. The topologial space (M(𝐸), 𝜏ℎ𝑘 ) is a Hausdorff space. 11 Remember that a subset 𝐴 in a vector lattice is complete if 𝑥 ⊥ 𝐴 implies 𝑥 = 0.

110 | 5 The space of maximal ideals of a vector lattice Proof. Let 𝐸 be an arbitrary (radical-free) vector lattice, and 𝑀1 , 𝑀2 ∈ M(𝐸) with 𝑀1 ≠ 𝑀2 . It is sufficient to find two elements 𝑥, 𝑦 ∈ 𝐸 such that 𝑥 ∈ 𝑀2 \ 𝑀1 , 𝑦 ∈ 𝑀1 \ 𝑀2 and 𝑥 ⊥ 𝑦. Indeed, if this is the case then 𝑀1 ∈ 𝐺𝑥 , 𝑀2 ∈ 𝐺𝑦 and according to Proposition 5.13 and Theorem 5.14, the sets 𝐺𝑥, 𝐺𝑦 are disjoint neighborhoods of 𝑀1 and 𝑀2 respectively. Now, in view of 𝑀1 ≠ 𝑀2 and the maximality of both 𝑀𝑖 , there are two positive nonzero elements 𝑧𝑖 ∈ 𝑀𝑖 (𝑖 = 1, 2) with 𝑧𝑖 ∉ 𝑀𝑗 for 𝑖 ≠ 𝑗, i. e., in both of the maximal ideals 𝑀𝑖 , 𝑖 = 1, 2 there are nonzero elements not belonging to the other¹² one. If the two elements are disjoint then we are finished. Otherwise we consider the elements 𝑥 = 𝑧1 − 𝑧1 ∧ 𝑧2 , and 𝑦 = 𝑧2 − 𝑧1 ∧ 𝑧2 , for which one has 𝑥 ∧ 𝑦 = (𝑧1 − 𝑧1 ∧ 𝑧2 ) ∧ (𝑧2 − 𝑧1 ∧ 𝑧2 ) = 𝑧1 ∧ 𝑧2 − 𝑧1 ∧ 𝑧2 = 0. So 𝑥 ⊥ 𝑦. Finally, the inequalities 𝑥 ≤ 𝑧1 and 𝑦 ≤ 𝑧2 imply 𝑥 ∈ 𝑀1 and 𝑦 ∈ 𝑀2 . In general, no stronger separation axiom can be expected to hold, since even for a vector lattice 𝐸 of type (Σ) the space (M(𝐸), 𝜏ℎ𝑘 ) might not be a regular topological space, as the next example shows. Example 5.17. A vector lattice 𝐸 of type (Σ), where the space (M(𝐸), 𝜏ℎ𝑘 ) is not regular. We supplement our knowledge about the Kaplansky vector lattice K, see Example 3.4, with an additional investigation of properties of its space M(K). It is convenient further on not to distinguish between a function from 𝐶([−2, 2]) and its restriction to 𝑇 = [−2, 2] \ {1, 12 , 13 , . . .}, and to agree that 𝐶([−2, 2]) ⊂ K. The symbol 1 stands for the function identical to 1 on [−2, 2]. Consider the discrete functionals 𝑓𝑡 corresponding to the points 𝑡 ∈ [−2, 2]. For 𝑡 ≠ 𝑛1 , 𝑛 ∈ ℕ these are the valuation functionals at 𝑡, i. e., 𝑓𝑡 (𝑥) = 𝑥(𝑡) 𝑥 ∈ K. For the points 𝑡 =

1 𝑛

the functional 𝑓𝑡 is defined by 𝑓1 (𝑥) = lim1 |𝑡 𝑛 − 1| 𝑥(𝑡), 𝑛

𝑡→ 𝑛

𝑥 ∈ K.

That way an injective map is produced from [−2, 2] into the set of all discrete functionals on K, and consequently also into M(K). Further on we identify each point 𝑡 ∈ [−2, 2] with the maximal ideal 𝑓𝑡−1 (0) ∈ M(K). Our next aim is to show that M(K) = [−2, 2]. Let 𝑀 be an arbitrary maximal ideal in K and 𝑓 its corresponding discrete functional on K. We have two alternatives: either a continuous function on [−2, 2] exists, on which 𝑓 does not vanish, or 𝑓 is identical to zero on 𝐶([−2, 2]). In the first case, obviously, 𝑓(1) > 0, where, without loss of generality, even

12 If any 𝑧 ∈ 𝑀2 also belongs to 𝑀1 , then due to the maximality of 𝑀2 , the relation 𝑀2 ⊂ 𝑀1 implies the contradictory equality 𝑀1 = 𝑀2 .

5.4 The Hausdorff property of M |

111

𝑓(1) = 1 can be assumed. The restriction 𝑓 ̃ of 𝑓 on 𝐶([−2, 2]) is a discrete functional on 𝐶([−2, 2]), such that there is some point 𝑡0 ∈ [−2, 2] with ̃ 𝑓(𝑥) = 𝑥(𝑡0 ),

𝑥 ∈ 𝐶([−2, 2]).

We establish 𝑡0 ≠ 1𝑛 , where 𝑛 ∈ ℕ. If there were 𝑡0 = 1𝑛 for some 𝑛 ∈ ℕ, then take the 1 function 𝑧(𝑡) = |𝑛𝑡−1| and put 𝑧𝑘 = 𝑧 ∧ 𝑘1 for all 𝑘 ∈ ℕ. Since 𝑧 ∈ 𝐸, one has ̃ ) = 𝑧 (𝑡 ) = 𝑘 𝑓(𝑧) ≥ 𝑓(𝑧𝑘 ) = 𝑓(𝑧 𝑘 𝑘 0

for all 𝑘 ∈ ℕ,

which is impossible. We now show 𝑓 = 𝑓𝑡0 . Let 0 < 𝑥 ∈ 𝐸. Since the function 𝑥 ∧ 𝑘1 belongs to 𝐶([−2, 2]), on one hand we have ̃ ∧ 𝑘1) = (𝑥 ∧ 𝑘1)(𝑡 ) = 𝑥(𝑡 ) ∧ 𝑘. 𝑓(𝑥 ∧ 𝑘1) = 𝑓(𝑥 0 0 On the other hand, 𝑓(𝑥 ∧ 𝑘1) = 𝑓(𝑥) ∧ 𝑓(𝑘1) = 𝑓(𝑥) ∧ 𝑘. Therefore, 𝑓(𝑥) ∧ 𝑘 = 𝑥(𝑡0 ) ∧ 𝑘. Since 𝑘 can be chosen large enough, we conclude 𝑓(𝑥) = 𝑥(𝑡0 ). In the second case, where the functional 𝑓 vanishes on 𝐶([−2, 2]), we define the functions 1 𝑥𝑚 (𝑡) = , 𝑡 ∈ 𝑇, 𝑚 ∈ ℕ. |𝑚𝑡 − 1| If 𝑚1 ≠ 𝑚2 then 𝑥𝑚1 ∧ 𝑥𝑚2 ∈ 𝐶([−2, 2]) and therefore, 𝑓(𝑥𝑚1 ) ∧ 𝑓(𝑥𝑚2 ) = 0. It is clear that the functional 𝑓 has to be different from zero on one of the functions 𝑥𝑚 , say on 𝑥𝑚0 . So we show 𝑓−1 (0) = 𝑓−1 1 (0), hence the functional 𝑓 is proportional to the 𝑚0

functional 𝑓 1 . Due to their maximality it is sufficient to prove the inclusion 𝑓−1 1 (0) ⊂ 𝑚0

𝑓−1 (0). Let 0 < 𝑥 ∈ 𝐸 with 𝑓 1 (𝑥) = 0. For each 𝜀 > 0 there is a 𝛿 > 0, such that

𝑚0

𝑚0

𝑥(𝑡) ≤ 𝜀 𝑥𝑚0 (𝑡) for 𝑡 ≠

1 , 𝑚0

𝑡 ∈ ( 𝑚1 − 𝛿, 0

1 𝑚0

+ 𝛿).

(5.3)

On the other hand, for some natural 𝑁 one has 𝑁

𝑥 ≤ 𝑐𝑥 𝑒𝑁 = 𝑐𝑥 ∑ 𝑥𝑘 + 𝑐𝑥 𝑥𝑚0 𝑘=1 𝑘=𝑚 ̸ 0

The function 𝑥𝑚0 is bounded outside the interval ( 𝑚1 − 𝛿, 0

󸀠

𝑐 there holds the inequality 𝑁

𝑥(𝑡) ≤ 𝑐𝑥 ∑ 𝑥𝑘 + 𝑐󸀠 , 𝑘=1 𝑘=𝑚 ̸ 0

1 𝑚0

+ 𝛿), so with some constant

112 | 5 The space of maximal ideals of a vector lattice which, together with (5.3), yields 𝑁

𝑥 ≤ 𝜀 𝑥𝑚0 + 𝑐󸀠 1 + 𝑐𝑥 ∑ 𝑥𝑘 . 𝑘=1 𝑘=𝑚 ̸ 0

After the functional 𝑓 is applied, we get 0 ≤ 𝑓(𝑥) ≤ 𝜀 𝑓(𝑥𝑚0 ), from where 𝑓(𝑥) = 0, or equivalently 𝑥 ∈ 𝑓−1 (0) follows immediately. Due to the already mentioned identificationof M(K) with the interval [−2, 2], we can now investigate the topology 𝜏ℎ𝑘 on [−2, 2] (and compare it with the induced topology from ℝ1 ). A neighborhood system of a point 𝑡 ≠ 0 in the topology 𝜏ℎ𝑘 is the system of usual open intervals (𝑡 − 𝛿, 𝑡 + 𝛿). According to Theorem 5.14 (2), a neighborhood system of the point 𝑡 = 0 with respect to the topology 𝜏ℎ𝑘 , consists of the sets 𝐺𝑧 which contain the point 𝑡 = 0, i. e., of the sets 𝐺𝑧 for 𝑧 ∈ 𝐸 with 𝑧(0) ≠ 0. Such functions are continuous at 𝑡 = 0, and therefore bounded on some set 𝑇 ∩ (−𝛿, 𝛿) for 𝛿 > 0. For a sufficiently large 𝑛 one has 𝑓1 (𝑧) = lim1 |𝑛𝑡 − 1| 𝑧(𝑡) = 0, 𝑛

𝑡→ 𝑛

which shows that the set (−𝛿, 𝛿) \ { 1𝑛 : 𝑛 ∈ ℕ} is a subset of 𝐺𝑧 . Conversely, each set (−𝛿, 𝛿) \ { 1𝑛 : 𝑛 ∈ ℕ} contains a set 𝐺𝑧 ∋ 0, provided 𝑧 is a continuous function which vanishes outside of (−𝛿, 𝛿), and satisfies 𝑧(0) ≠ 0. Finally, we consider the set 𝐹 = {1, 12 , 13 , . . .}, and show that 𝐹 and all its subsets are closed with respect to the topology 𝜏ℎ𝑘 . Let 𝐹󸀠 = { 𝑛1 , 𝑛1 , . . . , 𝑛1 , . . .} be an arbitrary subset of 𝐹. We will show that the set 𝑇 \ 𝐹󸀠 is open 1

2

𝑘

with respect to the topology 𝜏ℎ𝑘 . Each point 𝑡 ∈ 𝑇 \ 𝐹󸀠 possesses either a neighborhood of kind (𝑡 − 𝛿, 𝑡 + 𝛿) for sufficiently small 𝛿 > 0, or in case 𝑡 = 0, a neighborhood of kind (−𝛿, 𝛿) \ { 𝑛1 : 𝑛 ∈ ℕ}, which is disjoint to 𝐹󸀠 . The point 𝑡 = 0 and the set 𝐹 can not be separated by disjoint open sets. That is why the space M(K) is not regular. Example 5.18. The topology 𝜏ℎ𝑘 on (M(𝐸) for the vector lattice 𝐸 = 𝐸(𝑇) of Example 3.5 and its comparison with the topology of the real line. We continue our investigation of the space (M(𝐸), 𝜏ℎ𝑘 ) for the example which has already been dealt with in Section 3.1, where it was shown that Φ1 (𝐸) = {0} is possible. Remember that in this example 𝐸 = 𝐸(𝑇) was the vector lattice¹³ of type (Σ), consisting of all functions 𝑥 on 𝑇 = [0, 1] which are continuous on 𝑇 except a finite number of rational points, i. e., 𝑁 𝛼𝑚 𝑥(𝑡) = 𝛼(𝑡) + ∑ , 𝑚=1 |𝑡 − 𝑟𝑚 | where 𝛼(𝑡) is some continuous function on 𝑇, 𝛼1, 𝛼2 , . . . , 𝛼𝑚 are real numbers, 𝑁 = 𝑁(𝑥) ∈ ℕ, and 𝑟𝑚 ∈ ℚ ∩ 𝑇, i. e., rational numbers in 𝑇. Analogously, as in the previous example,

13 The functions 𝑒𝜈 (𝑡) = 1 + ∑𝜈𝑚=1

1 , |𝑡−𝑟𝑚 |

𝜈 = 1, 2, . . . belong to 𝐸 and satisfy the condition (Σ󸀠 ).

5.4 The Hausdorff property of M |

113

one can show that an arbitrary discrete functional on the 𝐸 is proportional to one of the following: 𝑓𝑡 (𝑥) = 𝑥(𝑡), if 𝑡 ∈ 𝑇 is irrational 𝑓𝑡 (𝑥) = lim |𝑠 − 𝑡| 𝑥(𝑠), if 𝑡 ∈ 𝑇 is rational. 𝑠→𝑡

One can conclude that the set of all maximal ideals in 𝐸 is in one-to-one correspondence to all points of 𝑇, and may therefore be identified, i. e., 𝑡 ∈ 𝑇 with 𝑓𝑡−1 (0), and so M(𝐸) = [0, 1] = 𝑇. We are now able to clarify the topology of the space (M(𝐸), 𝜏ℎ𝑘 ) and compare it with the usual topology 𝜏 induced on 𝑇 by the topology of the real line. The 𝜏-closure 𝜏 of a subset 𝐴 ⊂ 𝑇 will be denoted by 𝐴 and its 𝜏ℎ𝑘 -closure by ℎ(𝑘(𝐴)), respectively. By analogous consideration, as in the previous example, the following properties of 𝜏ℎ𝑘 on 𝑇 can be shown: (i) a subset of 𝑇 is 𝜏ℎ𝑘 -open if and only if it is a 𝜏-open set, where some rational numbers are deleted; (ii) if a set consists of rational points only, then it is 𝜏ℎ𝑘 -closed. This implies that the set of all irrational points of 𝑇 is 𝜏ℎ𝑘 -open; (iii) a subset of 𝑇 is 𝜏ℎ𝑘 -closed if it is the union of a 𝜏-closed set and a set of rational points: (iv) each 𝜏-closed set is also 𝜏ℎ𝑘 -closed. If a set 𝐴 ⊂ 𝑇 consists only of irrational points, then 𝑘(𝐴) = {𝑥 ∈ 𝐸(𝑇) : 𝑥(𝑡) = 0,

𝑡 ∈ 𝐴},

where 𝑘(𝐴) is the kernel of the set 𝐴, where the latter is regarded as a subset of M(𝐸). 𝜏 We will show that for such sets the relation 𝐴 = ℎ(𝑘(𝐴)) holds. a) Let 𝑠 ∈ 𝑇 be an irrational point which belongs to ℎ(𝑘(𝐴)), i. e., 𝑠 is a 𝜏ℎ𝑘 -closure point of 𝐴, and let 𝑥 ∈ 𝑘(𝐴) ∩ 𝐶(𝑇). We are going to show that 𝑥 ∈ 𝑓𝑠−1 (0), which, due to the irrationality of 𝑠, means 𝑥(𝑠) = 0. If it would be 𝑥(𝑠) ≠ 0, then 𝑠 ∈ 𝐺𝑥 = {𝑡 ∈ 𝑇: 𝑥 ∉ 𝑓𝑡−1 (0)}. Since 𝐺𝑥 is a 𝜏ℎ𝑘 -neighborhood of the 𝜏ℎ𝑘 -closure point 𝑠 of the set 𝐴, one has 𝐺𝑥 ∩ 𝐴 ≠ 0. For any point 𝑡0 ∈ 𝐺𝑥 ∩ 𝐴, due to its irrationality, one has, on one hand 𝑥 ∉ 𝑓𝑡−1 (0) = {𝑥󸀠 ∈ 𝐸(𝑇) : 𝑥󸀠 (𝑡0 ) = 0}, i. e., 𝑥(𝑡0 ) ≠ 0, and on the other hand, due to 0 𝑥 ∈ 𝑘(𝐴), one has 𝑥(𝑡0 ) = 0. Each continuous function 𝑥 ∈ 𝑘(𝐴) vanishes at the point 𝑠, 𝜏 which implies 𝑠 ∈ 𝐴 . 𝜏 Conversely, let 𝑠 be an irrational point which belongs to 𝐴 . If 𝑥 ∈ 𝑘(𝐴), then 𝑥(𝑡) = 0 for all 𝑡 ∈ 𝐴, and 𝑥 is continuous at 𝑠. Then 𝑥(𝑠) = 0, and consequently 𝑓𝑠−1 (0) ⊃ 𝑘(𝐴), i. e., 𝑠 ∈ ℎ(𝑘(𝐴)). b) Let 𝑠 ∈ 𝑇 be a rational point which belongs to ℎ(𝑘(𝐴)), i. e., 𝑓𝑠−1 (0) = { 𝑥 ∈ 𝐸(𝑇) : |𝑡 − 𝑠| 𝑥(𝑡) 󳨀→ 0} ⊃ 𝑘(𝐴). 𝑡→𝑠 𝑡∈𝐴

𝜏

Then each function of 𝑘(𝐴) is continuous at 𝑠. If there would be 𝑠 ∉ 𝐴 , then for 𝑥 ∈ 𝑥(𝑡) 𝑘(𝐴) ∩ 𝐶(𝑇) the function |𝑡−𝑠| belongs to 𝑘(𝐴), however it is discontinuous at the point 𝑠. 𝜏

It follows that 𝑠 ∈ 𝐴 .

114 | 5 The space of maximal ideals of a vector lattice 𝜏

Conversely, let 𝑠 be a rational point that belongs to 𝐴 . If 𝑥 ∈ 𝑘(𝐴), then 𝑥(𝑡) = 0 for all 𝑡 ∈ 𝐴, and the function 𝑥(𝑡) does not converge to ∞ as 𝑡 → 𝑠, and so 𝑥 is continuous at the point 𝑠. Consequently, |𝑡 − 𝑠| 𝑥(𝑡) 󳨀→ 0, 𝑡→𝑠

i. e., 𝑥 ∈ 𝑓𝑠−1 (0),

which shows 𝑓−1 (0) ⊃ 𝑘(𝐴) or, in other words, 𝑠 ∈ ℎ(𝑘(𝐴)). We are now able to add three more properties of the topology 𝜏ℎ𝑘 to the above list: 𝜏 (v) if a set 𝐴 ⊂ 𝑇 consists only of irrational points, then ℎ(𝑘(𝐴)) = 𝐴 ; (vi) since each set 𝐴 ⊂ 𝑇 can be decomposed into two parts (its rational one 𝐴 𝑟 and its irrational one 𝐴 𝑖 ), for the 𝜏ℎ𝑘 -closure of 𝐴 one has the simple relation 𝜏

ℎ(𝑘(𝐴)) = ℎ(𝑘(𝐴 𝑟 )) ∪ ℎ(𝑘(𝐴 𝑖 )) = 𝐴 𝑟 ∪ 𝐴𝑖 ; (vii) it is easy to see that an arbitrary subset of 𝑇 which contains a 𝜏ℎ𝑘 - interior point cannot be compact. From Example 5.17 we immediately see that not every prime ideal in a vector lattice is a maximal ideal. Example 5.19. A vector lattice 𝐸 of type (Σ), where some prime ideals are not maximal ideals in 𝐸. The description of the discrete functionals in the vector lattice K in Example 5.17 shows that for any 𝑛 ∈ ℕ, the set 𝑃𝑛 = {𝑥 ∈ K : 𝑥( 1𝑛 ) = 0} is an ideal, which is prime but not maximal. It is clear that 𝑃𝑛 ⊂ 𝑓−1 1 (0), and e. g., any continuous function 𝑧 ∈ K with 𝑛

𝑧( 1𝑛 ) ≠ 0 belongs to 𝑓−1 (0) but not to 𝑃𝑛. In Section 6.3 we continue the study of the space M(𝐸) for vector lattices of type (Σ).

6 Topological characterization of finite elements The finite, totally finite and selfmajorizing elements of an Archimedean vector lattice 𝐸 allow a nice topological characterization by means of special subsets of the space M(𝐸) of its maximal ideals. This obstacle will be used in the subsequent chapters, both for a deeper study of the collection of finite elements in vector lattices (see [90, 91]) [114], as well as for further development of the representation theory (see [89, 92]).

6.1 Topological characterization of finite, totally finite and selfmajorizing elements In the proofs of some theorems in subsequent sections, and also for the construction of several examples, it is convenient to use notions of convergent ultrafilters and the limit of functions along an ultrafilter in topological spaces. We provide some basic notions on filters and ultrafilters in order to facilitate further reading (see [31, § I.6– 7]). Let 𝑋 be some given nonempty set. A filter in 𝑋 is set F of subsets of 𝑋 with the following properties: (F1 ) any subset of 𝐴 ⊂ 𝑋 containing a set of F belongs to F; (F2 ) the intersection of any finite number of sets of F belongs to F; (F3 ) 0 ∉ F. Let F and F󸀠 be two filters in the same set 𝑋. We say that F󸀠 majorizes F if F ⊂ F󸀠 . An ultrafilter U in the set 𝑋 is a filter which is not majorized by any filter F in 𝑋, such that F ≠ U. Among all filters the ultrafilters are characterized as follows: for a filter F in 𝑋 there are equivalent (i) F is an ultrafilter in 𝑋, (ii) if 𝐴 ∪ 𝐵 ∈ F then either 𝐴 ∈ F or 𝐵 ∈ F. A subset B of a filter F in 𝑋 is the basis of F, if F is the collection of all subsets in 𝑋, each of them contains some set of B. A necessary and sufficient condition for a subset B of a filter F to be a basis is that each set of F contains some set of B. It is easy to see that for a map 𝑓 from the set 𝑋 into the set 𝑋󸀠 , the image 𝑓(B) of basis B in 𝑋 is a basis in 𝑋󸀠 . Now let 𝑋 be a topological space. Then for any point 𝑥 ∈ 𝑋, the collection N(𝑥) of all neighborhoods of 𝑥 is a filter in 𝑋. A filter F in the space 𝑋 is said to converge to 𝑥 if F majorizes the filter N(𝑥), i. e., N(𝑥) ⊂ F. In this case, the point 𝑥 is called limit of the filter F. If 𝐴 ⊂ 𝑋 and 𝑥 ∈ 𝐴, then the trace of N(𝑥) on 𝐴 is a filter in 𝐴. Since each filter is majorized by an ultrafilter, each closure point 𝑥 ∈ 𝐴 is the limit of some ultrafilter in 𝐴.

116 | 6 Topological characterization of finite elements The point 𝑥 is called limit of the basis B (of a filter) in 𝑋 if the filter with basis B converges to 𝑥. It is well known that a filter can have only one limit in a Hausdorff space . Definition 6.1. Let 𝑓 : 𝑋 → ℝ be a function on the set 𝑋 into (the topological space) ℝ and F a filter in 𝑋. The number 𝛼 ∈ ℝ is called limit of 𝑓 along the filter F if the basis of the filter 𝑓(F) converges to 𝛼. This is denoted by lim 𝑓(𝑥) = 𝛼 or 𝑓(𝑥) 󳨀→ 𝛼. F

F

If 𝑋 is a topological space, a function 𝑓 is continuous at the point 𝑥0 ∈ 𝑋 if and only if 𝑓(𝑥) 󳨀→ 𝑓(𝑥0 ). N(𝑥0 )

In this section, let 𝐸0 be an ideal in the vector lattice 𝐸, and, as usual, 𝐺(𝐸0 ) = {𝑀 ∈ M(𝐸) : ∃𝑥 ∈ 𝐸0 such that 𝑥 ∉ 𝑀}.

6.1.1 The canonical map and the conditional representation The investigation of maximal ideals both in a vector lattice 𝐸, and in one of its ideals (if the latter is considered a vector lattice on its own), yields the notion of the canonical map. Definition 6.2. Let 𝐸 be a vector lattice and 𝐸0 an ideal of 𝐸. The map 𝜔 : 𝐺(𝐸0 ) → M(𝐸0 ), defined by the formula 𝜔(𝑀) = 𝑀 ∩ 𝐸0 ,

𝑀 ∈ 𝐺(𝐸0 ),

is called the canonical map, where 𝐺(𝐸0 ) = ⋃𝑥∈𝐸0 𝐺𝑥 . Proposition 6.3. The canonical map is a homeomorphism from 𝐺(𝐸0 ) onto 𝜔(𝐺(𝐸0 )) ⊂ M(𝐸0 )). Proof. We first show the injectivity of the map 𝜔. For 𝑀1 , 𝑀2 ∈ 𝐺(𝐸0 ), 𝑀1 ≠ 𝑀2 there exist elements 0 < 𝑥 ∈ 𝐸 with 𝑥 ∈ 𝑀1 , 𝑥 ∉ 𝑀2 , and 0 < 𝑧 ∈ 𝐸0 with 𝑧 ∉ 𝑀2 . One has 𝑥 ∧ 𝑧 ∈ 𝑀1 , 𝑥 ∧ 𝑧 ∉ 𝑀2 , 𝑥 ∧ 𝑧 ∈ 𝐸0 , and therefore 𝑀1 ∩ 𝐸0 ≠ 𝑀2 ∩ 𝐸0 , i. e., 𝜔(𝑀1 ) ≠ 𝜔(𝑀2 ). Now we show that 𝜔 maps a closure point of any subset at a closure point. Let M ⊂ 𝐺(𝐸0 ), 𝑀0 ∈ 𝐺(𝐸0 ), and 𝑀0 ∈ M (the closure of M). From 𝑀0 ⊃ ⋂𝑀∈M 𝑀 it follows that 𝑀0 ∩ 𝐸0 ⊃ ⋂𝑀∈M (𝑀 ∩ 𝐸0 ). The latter inclusion means that 𝜔(𝑀0 ) lies in the closure of the set 𝜔(M). Finally, we prove that 𝜔−1 also maps any closure point at a closure point. For that, let 𝐵 ⊂ 𝜔(𝐺(𝐸0 )), 𝑁0 ∈ 𝜔(𝐺(𝐸0 )), 𝑁0 ∈ 𝐵. We show that the maximal ideal 𝑀0 = 𝜔−1 (𝑁0 ) is a closure point of the set M = 𝜔−1 (𝐵). Due to the assumptions, one has 𝑁0 = 𝑀0 ∩ 𝐸0 ⊃ ⋂𝑀∈M (𝑀 ∩ 𝐸0 ). We have to show 𝑀0 ⊃ ⋂𝑀∈M 𝑀. Let 0 ≤ 𝑥 ∈ ⋂𝑀∈M 𝑀. If 0 ≤ 𝑧 ∈ 𝐸0 and 𝑧 ∉ 𝑀0 , then 𝑥 ∧ 𝑧 ∈ 𝑀 for each 𝑀 ∈ M, and 𝑥 ∧ 𝑧 ∈ 𝐸0 . Therefore 𝑥 ∧ 𝑧 ∈ 𝑀0 ∩ 𝐸0 ⊂ 𝑀0 . Since 𝑧 ∉ 𝑀0 there follows 𝑥 ∈ 𝑀0 .

6.1 Topological characterization of finite, totally finite and selfmajorizing elements

| 117

In order to establish the main results of this section we need some facts concerning maximal ideals in a vector lattice 𝐸 and in its ideals. Proposition 6.4. Let 𝐸0 be an ideal of the vector lattice 𝐸. If a finite element exists in 𝐸 which does not belong to the maximal ideal 𝑁 ∈ M(𝐸0 ) and has a majorant in 𝐸0 , then 𝑁 ∈ 𝜔(𝐺(𝐸0 )), i. e., 𝑁 can be uniquely extended to a maximal ideal 𝑀 in 𝐸 such that 𝑀 ∩ 𝐸0 = 𝑁. Proof. Let 𝑥0 be a finite element with 𝑥0 ∉ 𝑁 and 𝑧 ∈ 𝐸0 its 𝐸-majorant. We may assume 𝑥0 > 0. Let 𝑔 be a discrete functional corresponding to 𝑁, i. e., 𝑔−1 (0) = 𝑁. Put 𝑓0 (𝑥) = sup 𝑔(𝑥 ∧ 𝛼𝑥0) 𝛼>0

for 𝑥 ∈ 𝐸+ .

Then 𝑓0 (𝑥) < +∞ , since 𝑥 ∧ 𝛼𝑥0 ≤ 𝜆 𝑥𝑧 for some 𝜆 𝑥 > 0. It is clear that 𝑥 ≥ 𝑦 ≥ 0 implies 𝑓0 (𝑥) ≥ 𝑓0 (𝑦) ≥ 0. By the formula 𝑓(𝑥) = 𝑓0 (𝑥+ ) − 𝑓0 (𝑥− ),

where 𝑥 ∈ 𝐸, 𝑥 = 𝑥+ − 𝑥− ,

(6.1)

the functional 𝑓0 is extended from 𝐸+ onto 𝐸. It will be shown that 𝑓 is a discrete functional which coincides on 𝐸0 with 𝑔. Obviously, the functional 𝑓 is positive and homogeneous for nonnegative scalars, i. e., 𝑓(𝜆𝑥) = 𝜆𝑓(𝑥) for 𝜆 ≥ 0. From (−𝑥)+ = (−𝑥) ∨ 0 = 𝑥− and (−𝑥)− = 𝑥 ∨ 0 = 𝑥+ we get 𝑓(−𝑥) = 𝑓0 (𝑥− ) − 𝑓0 (𝑥+ ) = −𝑓(𝑥), thus 𝑓 is homogeneous. Its additivity will follow according to [120, Lemma VIII.1.1], if (6.2)

𝑓0 (𝑥 + 𝑦) = 𝑓0 (𝑥) + 𝑓0 (𝑦)

is established for 𝑥, 𝑦 ≥ 0. The inequality 𝑔((𝑥 + 𝑦) ∧ 𝛼𝑥0 ) ≤ 𝑔(𝑥 ∧ 𝛼𝑥0 ) + 𝑔(𝑦 ∧ 𝛼𝑥0 ) implies on one hand, 𝑓0 (𝑥 + 𝑦) ≤ 𝑓0 (𝑥) + 𝑓0 (𝑦). (6.3) On the other hand, for arbitrary real numbers 𝛼, 𝛽 > 0, one has the relation 𝑔(𝑥 ∧ 𝛼𝑥0 ) + 𝑔(𝑦 ∧ 𝛽𝑥0 ) = 𝑔((𝑥 + 𝑦 ∧ 𝛽𝑥0 ) ∧ (𝛼𝑥0 + 𝑦 ∧ 𝛽𝑥0 )) = 𝑔((𝑥 + 𝑦) ∧ (𝑥 + 𝛽𝑥0 ) ∧ (𝑦 + 𝛼𝑥0) ∧ (𝛼𝑥0 + 𝛽𝑥0 )) ≤ 𝑔((𝑥 + 𝑦) ∧ (𝛼 + 𝛽)𝑥0 )) ≤ 𝑓0 (𝑥 + 𝑦), from where 𝑓0 (𝑥) + 𝑓0 (𝑦) ≤ 𝑓0 (𝑥 + 𝑦) follows. Together with (6.3), we then have (6.2). Now we show first that for 𝑥, 𝑦 ≥ 0 the following two equations hold: 𝑓0 (𝑥 ∧ 𝑦) = 𝑓0 (𝑥) ∧ 𝑓0 (𝑦) and 𝑓0 (𝑥 ∨ 𝑦) = 𝑓0 (𝑥) ∨ 𝑓0 (𝑦).

(6.4)

For the first one we have 𝑓0 (𝑥 ∧ 𝑦) = sup 𝑔(𝑥 ∧ 𝑦 ∧ 𝛼𝑥0 ) = sup 𝑔(𝑥 ∧ 𝑦 ∧ 𝛼𝑥0 ∧ 𝛽𝑥0 ) 𝛼>0

𝛼,𝛽>0

= sup (𝑔(𝑥 ∧ 𝛼𝑥0) ∧ 𝑔(𝑦 ∧ 𝛽𝑥0 )) = ( sup 𝑔(𝑥 ∧ 𝛼𝑥0 )) ∧ ( sup 𝑔(𝑦 ∧ 𝛽𝑥0 )) 𝛼,𝛽>0

= 𝑓0 (𝑥) ∧ 𝑓0 (𝑦).

𝛼>0

𝛽>0

118 | 6 Topological characterization of finite elements The second one is proved as follows. Due to 𝑥, 𝑦 ≤ 𝑥∨𝑦, there is 𝑓0 (𝑥)∨𝑓0 (𝑦) ≤ 𝑓0 (𝑥∨𝑦). Further 𝑓0 (𝑥 ∨ 𝑦) = sup 𝑔((𝑥 ∨ 𝑦) ∧ 𝛼𝑥0 ) = sup 𝑔((𝑥 ∧ 𝛼𝑥0 ) ∨ (𝑦 ∧ 𝛼𝑥0 )) 𝛼>0

𝛼>0

= sup (𝑔(𝑥 ∧ 𝛼𝑥0) ∨ 𝑔(𝑦 ∧ 𝛼𝑥0 )) ≤ 𝑓0 (𝑥) ∨ 𝑓0 (𝑦). 𝛼>0

In order to show that the functional 𝑓 is discrete, consider arbitrary elements 𝑥, 𝑦 ∈ 𝐸 and denote 𝑢 = (𝑥 ∨ 𝑦)+ , 𝑣 = (𝑥 ∧ 𝑦)− . Observe that 𝑢 = 𝑥+ ∧ 𝑦+ , 𝑣 = 𝑥− ∨ 𝑦− and 𝑢 ∧ 𝑣 = 0. Since 𝑓0 (𝑢) ∧ 𝑓0 (𝑣) = 𝑓0 (𝑢 ∧ 𝑣) = 𝑓0 (0) = 0, one of the numbers 𝑓0 (𝑢) or 𝑓0 (𝑣) is equal to zero. a) If 𝑓0 (𝑢) = 0, then because of 𝑢 = 𝑥+ ∧ 𝑦+ one of the numbers 𝑓0 (𝑥+ ) or 𝑓0 (𝑦+ ) is zero. Suppose 𝑓0 (𝑥+) = 0. If now 𝑓0 (𝑦+ ) is also zero, then 𝑓(𝑥) = −𝑓(𝑥− ), and 𝑓(𝑦) = −𝑓(𝑦− ), and therefore 𝑓(𝑥 ∧ 𝑦) = −𝑓0 (𝑣) = −𝑓0 (𝑥− ∨ 𝑦− ) = −(𝑓0 (𝑥− ) ∨ 𝑓0 (𝑦− )) = ( − 𝑓0 (𝑥−)) ∧ ( − 𝑓0 (𝑦− )) = 𝑓(𝑥) ∧ 𝑓(𝑦). If, however, 𝑓0 (𝑦+ ) > 0, then the equations 𝑓0 (𝑦+ ) ∧ 𝑓0 (𝑦− ) = 𝑓0 (𝑦+ ∧ 𝑦− ) = 𝑓0 (0) = 0 imply 𝑓0 (𝑦− ) = 0. In this case 𝑓(𝑥) ≤ 0, and 𝑓(𝑦) > 0, which yields 𝑓(𝑥) ∧ 𝑓(𝑦) = 𝑓(𝑥). Moreover, since 𝑓(𝑥 ∧ 𝑦) = −𝑓0 (𝑥− ∨ 𝑦− ) = −(𝑓0 (𝑥− ) ∨ 𝑓0 (𝑦− )) = −𝑓0 (𝑥− ) = 𝑓(𝑥), one has 𝑓(𝑥 ∧ 𝑦) = 𝑓(𝑥) ∧ 𝑓(𝑦). b) Let 𝑓0 (𝑣) = 0. Because of 𝑣 = 𝑥− ∨ 𝑦− there is 𝑥− ≤ 𝑣, 𝑦− ≤ 𝑣 and 𝑓0 (𝑥−) = 𝑓0 (𝑦− ) = 0. Therefore 𝑓(𝑥) = 𝑓0 (𝑥+ ) and 𝑓(𝑦) = 𝑓0 (𝑦+ ) hold. This yields 𝑓(𝑥 ∧ 𝑦) = 𝑓0 (𝑥+ ∧ 𝑦+ ) = 𝑓0 (𝑥+ ) ∧ 𝑓0 (𝑦+ ) = 𝑓(𝑥) ∧ 𝑓(𝑦). Finally, we show that 𝑓 is the extension of the functional 𝑔. Indeed, let 0 ≤ 𝑥 ∈ 𝐸0 . Since, 𝑥0 ∉ 𝑁 one has 𝑔(𝑥0 ) > 0, which results in the equation 𝑔(𝑥 ∧ 𝛼𝑥0) = 𝑔(𝑥), whenever 𝛼 is sufficiently large. It follows 𝑓(𝑥) = sup𝛼>0 𝑔(𝑥 ∧ 𝛼𝑥0 ) = 𝑔(𝑥). The hyperplane 𝑓−1 (0) is now the required maximal ideal 𝑀 in 𝐸. The uniqueness of 𝑀 is a consequence of the injectivity of the map 𝜔. Corollary 6.5. If for some vector lattice 𝐸 one has 𝐸 = Φ1 (𝐸), then 𝐸 is radical-free. Proof. Let 0 ≠ 𝑥0 ∈ 𝐸 and let 𝑧 be an 𝐸-majorant of 𝑥0 . The element 𝑥0 belongs to the Riesz subspace 𝐸𝑧 = {𝑥 ∈ 𝐸 : |𝑥| ≤ 𝑐𝑥 𝑧}, which is radical-free due to possessing the element 𝑧 as an order unit (see Remark on p. 105). Consequently, a maximal ideal 𝑁 ∈ M(𝐼𝑧 ) exists, to which 𝑥0 does not belong. According to the previous proposition, the maximal ideal 𝑁 can be extended to a maximal ideal 𝑀 ∈ M(𝐸). Then the element 𝑥0 does not belong to 𝑀, thus 𝑅(𝐸) = {0}.

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| 119

Proposition 6.6. Let 𝑍 be a vector lattice of bounded elements (with the order unit 1), and 𝑍 its norm completion with respect to the 𝑢-norm¹ ‖𝑧‖1 = inf{𝜆 : |𝑧| ≤ 𝜆1},

𝑧 ∈ 𝑍.

Then the following statements hold: (a) let ℎ𝑛, ℎ0 be discrete functionals on 𝑍 and 𝑔𝑛, 𝑔0 (𝑛 = 1, . . .) their extensions on 𝑍. ∞ ∞ −1 −1 −1 Then ℎ−1 0 (0) ⊃ ⋂𝑛=1 ℎ𝑛 (0) implies 𝑔0 (0) ⊃ ⋂𝑛=1 𝑔𝑛 (0); (b) let 𝑥, 𝑦 ∈ 𝑍 be arbitrary elements and (𝜆 𝑘 )𝑘∈ℕ a sequence of positive numbers with 𝜆 𝑘 → ∞. Then the sequence (𝑧𝑛)𝑛∈ℕ with 𝑧𝑛 = ⋀𝑛𝑘=1 |𝑥 − 𝜆1 𝑦| is a Cauchy sequence 𝑘 with respect to the norm in 𝑍. Proof. By the Theorem of S. Kakutani, H. F. Bohnenblust, M. G. Krein, and S. G. Krein on the representation of Banach lattices with unit (see Theorem 2.15), the space 𝑍 is isometrically Riesz isomorphic to 𝐶(𝑄) for some compact space 𝑄, where the isomorphism is chosen such that the image of the element 1 is the identity function on 𝑄. The vector lattice 𝑍 is isomorphic to some dense linear sublattice of 𝐶(𝑄). Further on, we identify 𝑍 with 𝐶(𝑄) and consider 𝑍 a Riesz subspace of 𝐶(𝑄). (a) For the discrete functionals 𝑔𝑛 , (𝑛 = 0, 1, 2, . . .), there are points 𝑠𝑛 ∈ 𝑄 such that 𝑔𝑛−1 (0) = 𝛿𝑠−1 (0) for 𝑛 = 0, 1, 2, . . ., where as usual the symbol 𝛿𝑠 denotes the eval𝑛 uation functional on 𝐶(𝑄), assigning to each function its value at the point 𝑠. The as∞ −1 sumption ℎ−1 0 (0) ⊃ ⋂𝑛=1 ℎ𝑛 (0) can be reformulated as follows: if 𝑧 is an element of 𝑍 which satisfies 𝑧(𝑠𝑛) = 0 for all 𝑛 = 1, 2 . . ., then also 𝑧(𝑠0 ) = 0. Since 𝑍 is dense in 𝐶(𝑄) ∞ we get that 𝑠0 is a closure point of the set {𝑠1 , 𝑠2 , . . .}. This means 𝛿𝑠−1 (0) ⊃ ⋂𝑛=1 𝛿𝑠−1 (0), 0 𝑛 ∞ −1 −1 i. e., 𝑔0 (0) ⊃ ⋂𝑛=1 𝑔𝑛 (0). (b) We fix an arbitrary 𝜀 > 0 and take a number 𝑁, such that for all 𝑛 ≥ 𝑁 the inequality 𝜆 𝑛 > 2𝜀 holds. Then for all 𝑛 ≥ 𝑁 one has the relations 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 |𝑥| − 2𝜀 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 = (𝑥 − 2𝜀 󵄨󵄨󵄨𝑦󵄨󵄨󵄨) ∨ (−𝑥 − 2𝜀 󵄨󵄨󵄨𝑦󵄨󵄨󵄨) ≤ (𝑥 − 𝜆1 𝑦) ∨ (−𝑥 + 𝜆1 𝑦) = 󵄨󵄨󵄨𝑥 − 𝜆1 𝑦󵄨󵄨󵄨 𝑛 𝑛 𝑛 and

󵄨󵄨 1 󵄨 1 󵄨 󵄨 𝜀 󵄨 󵄨 󵄨󵄨𝑥 − 𝜆 𝑦󵄨󵄨󵄨 ≤ |𝑥| + 𝜆 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ≤ |𝑥| + 2 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 . 𝑛 𝑛 From 𝑧1 ≥ 𝑧2 ≥ ⋅ ⋅ ⋅ we get for 𝑛 ≥ 𝑁 and any natural number 𝑝 the estimates 󵄨󵄨 󵄨 󵄨󵄨𝑧𝑛+𝑝 − 𝑧𝑛 󵄨󵄨󵄨 = 𝑧𝑛−1 ∧ 󵄨󵄨󵄨󵄨𝑥 − 󵄨 󵄨

1 󵄨󵄨 𝑦󵄨 𝜆𝑛 󵄨

𝑛+𝑝

󵄨 − 𝑧𝑛−1 ∧ ⋀ 󵄨󵄨󵄨𝑥 − 𝑘=𝑛

1 󵄨󵄨 𝑦󵄨 𝜆𝑘 󵄨

󵄨 󵄨 󵄨 󵄨 ≤ 𝑧𝑛−1 ∧ ( |𝑥| + 2𝜀 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ) − 𝑧𝑛−1 ∧ ( |𝑥| − 2𝜀 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 = 𝜀 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 + (𝑧𝑛−1 − 𝜀 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ) ∧ ( |𝑥| − 2𝜀 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ) − 𝑧𝑛−1 ∧ ( |𝑥| −

𝜀 2

󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨𝑦󵄨󵄨 ) ≤ 𝜀 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 .

Definition 6.7. For a fixed vector 𝑧 ∈ 𝐸, 𝑧 ≥ 0 and an arbitrary vector 𝑥 ∈ 𝐸, the conditional representation of 𝑥 (with respect to 𝑧) is defined by ̃ 𝑥(𝑀) := 𝑓(𝑥),

1 I. e., the 𝑢-norm for 𝑢 = 1.

where 𝑀 = 𝑓−1 (0), 𝑀 ∈ 𝐺𝑧 ,

𝑓(𝑧) = 1.

120 | 6 Topological characterization of finite elements ̃ ̃ Due to the positivity of any discrete functional, 𝑥 ≤ 𝑦 implies 𝑥(𝑀) ≤ 𝑦(𝑀) for all 𝑀 ∈ 𝐺𝑧 . The next result will be used in the proof of Theorem 6.10 Proposition 6.8. Let 𝑧 ∈ 𝐸, 𝑧 ≥ 0 and 𝐾 a compact set of M(𝐸), which is a subset of 𝐺𝑧 . Then the set 𝐷 = {𝑓 ∈ Δ(𝐸) : 𝑓−1 (0) ∈ 𝐾, 𝑓(𝑧) = 1} is weakly bounded, i. e., the conditional representation (with respect to 𝑧) of any vector 𝑥 ∈ 𝐸 is bounded on 𝐾. Proof. It is sufficient to prove that for each 𝑥 ∈ 𝐸 the value sup 𝑓(|𝑥|) 𝑓∈𝐷

󵄨 󵄨 is finite. By assuming the contrary there is an element 𝑦 ∈ 𝐸 with sup𝑓∈𝐷 𝑓(󵄨󵄨󵄨𝑦󵄨󵄨󵄨) = +∞. Therefore a sequence of functionals 𝑓𝑛 ∈ 𝐷 (𝑛 = 1, 2, . . .) exists such that 󵄨 󵄨 󵄨 󵄨 𝑓𝑛 (󵄨󵄨󵄨𝑦󵄨󵄨󵄨) 󳨀→ +∞. Denote the maximal ideals 𝑓𝑛−1 (0) by 𝑀𝑛 , and the numbers 𝑓𝑛 (󵄨󵄨󵄨𝑦󵄨󵄨󵄨) by 𝑛→∞ 𝜆 𝑛. The closed sets 𝐹𝑘 = {𝑀𝑘 , 𝑀𝑘+1 , . . .} belong to the set 𝐾 which, as a compact subset of the Hausdorff space M(𝐸), itself is ∞ closed. Since the system (𝐹𝑘 )𝑘∈ℕ has the finite intersection property², one has ⋂𝑘=1 𝐹𝑘 ≠ ∞ 0. Let 𝑀0 ∈ ⋂𝑘=1 𝐹𝑘 and 𝑓0 be the discrete functional corresponding to the maximal ideal 𝑀0 with 𝑓0 (𝑧) = 1. From 𝑀0 ∈ 𝐹𝑘 for any 𝑘 = 1, 2, . . . it follows that 𝑀0 ⊃ ⋂∞ 𝑛=𝑘 𝑀𝑛 . 󵄨󵄨 1 1 1 󵄨󵄨 Due to 𝑓𝑛 (𝑧 − 𝜆 𝑦) = 0, one has 𝑧 − 𝜆 𝑦 ∈ 𝑀𝑛 , which implies 󵄨󵄨𝑧 − 𝜆 𝑦󵄨󵄨 ∈ 𝑀𝑛 . Fix 𝑘0 such 𝑛 𝑛 𝑛 󵄨 󵄨 that 󵄨󵄨󵄨 𝜆1 𝑓0 (𝑦)󵄨󵄨󵄨 ≤ 12 for all 𝑛 ≥ 𝑘0 . Without loss of generality we assume 𝑘0 = 1. Denote 𝑛 𝑢=𝑧+

1 𝜆1

󵄨󵄨 󵄨󵄨 󵄨󵄨𝑦󵄨󵄨

and 𝑍 = 𝐼𝑢 = {𝑥 ∈ 𝐸 : |𝑥| ≤ 𝑐𝑥 𝑢}.

Denote by ℎ𝑛 (𝑛 = 0, 1, 2, . . .) the discrete functional which is the restriction of the functional 𝑓𝑛 on 𝑍. Let 𝑔𝑛 be the extension of the functional ℎ𝑛 onto the norm completion 𝑍 of the space 𝑍, when 𝑍 is considered with its 𝑢-norm. Clearly all elements 󵄨󵄨 1 󵄨󵄨 𝑧𝑚 = ⋀𝑚 𝑘=1 󵄨󵄨𝑧 − 𝜆 𝑦󵄨󵄨 belong to 𝑀𝑗 for all 𝑚 ≥ 𝑗. According to Proposition 6.6, the 𝑛

sequence (𝑧𝑚 )𝑚∈ℕ converges in 𝑍 to some element 𝑧0 . For fixed 𝑛 one has 𝑔𝑛(𝑧0 ) = lim 𝑔 (𝑧 ) = lim 𝑓𝑛 (𝑧𝑚 ) = 0. On one hand, again by Proposition 6.6, 𝑔0 (𝑧0 ) = 0. On 𝑚→∞ 𝑛 𝑚 𝑚→∞ the other hand the estimations 𝑚

󵄨 𝑔0 (𝑧0 ) = lim 𝑔0 (𝑧𝑚 ) = lim 𝑓0 (𝑧𝑚 ) = lim 𝑓0 ( ⋀ 󵄨󵄨󵄨𝑧 − 𝑚→∞

𝑚→∞

𝑚

󵄨 = lim ⋀ 󵄨󵄨󵄨𝑓0 (𝑧) − 𝑚→∞

𝑘=1

𝑚→∞

𝑘=1

1 󵄨󵄨 𝑦󵄨) 𝜆𝑘 󵄨

𝑚 1 󵄨󵄨 1 𝑓 (𝑦) ⋀ (1 − 12 ) = > 0 ≥ lim 󵄨 0 󵄨 𝜆𝑘 𝑚→∞ 2 𝑘=1

2 I. e., each finite subsystem has a nonempty intersection.

6.1 Topological characterization of finite, totally finite and selfmajorizing elements

| 121

hold. The contradiction reached finishes the proof of the boundedness of the set {𝑓(|𝑥|) : 𝑓 ∈ 𝐷} for each 𝑥 ∈ 𝐸. Proposition 6.9. The conditional representation of each vector 𝑥 ∈ 𝐼𝑧 = {𝑥 ∈ 𝐸 : ∃𝜆, |𝑥| ≤ 𝜆𝑧} is a continuous function on 𝐺𝑧 . Proof. Let 𝑖 be the isomorphic embedding of 𝐼𝑧 into the vector lattice 𝐶(M(𝐸)), such that 𝑖(𝑧) is the function identical to 1 on M(𝐸), and let 𝜔 be the canonical map from ̃ 𝐺𝑧 into M(𝐼𝑧 ). Then for 𝑥 ∈ 𝐼𝑧 and 𝑀 ∈ 𝐺𝑧 one has 𝑥(𝑀) = (𝑖(𝑥))(𝜔(𝑀)). Since 𝑖(𝑥) ∈ ̃ 𝐶(M(𝐸)) and the map 𝜔, by Proposition 6.3, is continuous on 𝐺𝑧 , the function 𝑥(𝑀) is continuous on 𝐺𝑧 .

6.1.2 Topological characterization of finite elements Further on, we consider Archimedean vector lattices which are assumed to be radicalfree. Theorem 6.10 (Topological characterization of finite elements). For a Archimedean vector lattice 𝐸 the following assertions hold: (1) 𝜑 ∈ Φ1 (𝐸) ⇐⇒ suppM (𝜑) is compact in (M(𝐸), 𝜏ℎ𝑘 ); (2) 𝑧 ∈ 𝐸 is an 𝐸-majorant of the finite element 𝜑 ⇐⇒ suppM (𝜑) ⊂ 𝐺𝑧 .

radical-free

Proof. Statements (1) and (2) will be proved simultaneously. For an arbitrary finite element 𝜑 and one of its 𝐸-majorants, say 𝑧, we show the compactness of the set suppM (𝜑) = 𝐺𝜑 and the inclusion suppM (𝜑) ⊂ 𝐺𝑧 . First of all, the inclusion 𝐺𝜑 ⊂ 𝐺𝑧 is clear. Since the space M(𝐼𝑧 ) is compact, and the canonical map 𝜔, according to Proposition 6.3, is a homeomorphism from 𝐺(𝐼𝑧 ) = 𝐺𝑧 onto 𝜔(𝐺𝑧 ) ⊂ M(𝐼𝑧 ), it is sufficient to establish the following two assertions: (a) 𝜔(𝐺𝜑 ) = 𝐺󸀠 , where 𝐺󸀠 = {𝑁 ∈ M(𝐼𝑧 ) : 𝜑 ∉ 𝑁}; (b) 𝐺󸀠 ⊂ 𝜔(𝐺𝑧 ), where 𝐺󸀠 denotes the closure of 𝐺󸀠 in the space M(𝐼𝑧 ). The assertion (a) follows immediately from Proposition 6.4, for 𝐸0 = 𝐼𝑧 . We move on to the proof of (b). Let 𝑁0 be a maximal ideal in 𝐼𝑧 , such that 𝑁0 ∈ 𝐺󸀠 , and U an ultrafilter in 𝐺󸀠 , which converges to 𝑁0 . According to Proposition 6.9, the conditional representation 𝑥̃̃ (with respect to 𝑧) of the element 𝑥 ∈ 𝐼𝑧 is a continuous function on M(𝐼𝑧 ), since 𝜔(𝐺𝑧 ) = {𝑁 = 𝑀∩𝐼𝑧 : 𝑀 ∈ M(𝐸), 𝑧 ∉ 𝑀}, i. e., 𝜔(𝐺𝑧 ) = M(𝐼𝑧 ). Therefore one has ̃̃ ̃̃ 0 ), whenever 𝑥 ∈ 𝐼𝑧 . 𝑥(𝑁) 󳨀→ 𝑥(𝑁 (6.5) U

Besides, for 𝑀 ∈ 𝐺𝑧 ⊂ M(𝐸) and 𝑥 ∈ 𝐼𝑧 the equation ̃̃ ̃ 𝑥(𝑀) = 𝑥(𝜔(𝑀))

(6.6)

122 | 6 Topological characterization of finite elements holds, where 𝑥̃ denotes the conditional representation of 𝑥 (as an element of 𝐸) with respect to 𝑧. Since the element 𝜑 is finite for any 𝑥 ∈ 𝐸 a number 𝑐𝑥 > 0 exists such that |𝑥| ∧ 󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨𝜑󵄨󵄨 ≤ 𝑐𝑥 𝑧 holds for all 𝑛 ∈ ℕ, which yields 󵄨 ̃ 󵄨󵄨 ̃ ∧ 𝑛 󵄨󵄨󵄨𝜑(𝑀) |𝑥(𝑀)| 󵄨󵄨 ≤ 𝑐𝑥

𝑀 ∈ M(𝐸).

̃ ̃ ≠ 0, so |𝑥(𝑀)| ≤ 𝑐𝑥 for all 𝑀 ∈ 𝐺𝜑 . Therefore the finite limit On 𝐺𝜑 one has³ 𝜑(𝑀) ̃ exists, where U∗ denotes the image of the ultrafilter U under the map 𝜔−1 . lim∗ 𝑥(𝑀) 𝑀∈U

̃ = 0} is a maximal ideal in 𝐸, which does Obviously, the set 𝑀0 = {𝑥 ∈ 𝐸 : lim∗ 𝑥(𝑀) 𝑀∈U

not contain the element 𝑧. Due to (6.5) and (6.6), the intersection of 𝑀0 with the ideal 𝐼𝑧 is exactly the set 𝑁0 . Therefore, 𝑁0 = 𝜔(𝑀0 ) ∈ 𝜔(𝐺𝑧 ). This proves the assertion (b), and hence the necessity of (1) and (2) of the theorem. Inversely, let the abstract support suppM (𝜑) of the element 𝜑 be a compact set such that suppM (𝜑) ⊂ 𝐺𝑧 , 𝑧 ∈ 𝐸, 𝑧 ≥ 0 (the existence of such 𝑧 is guaranteed by Theorem 5.14 (3)). We now show that the element 𝜑 is finite and 𝑧 is one of its 𝐸-majorants. In view of the totality⁴ of all discrete functionals on 𝐸, it suffices to check whether the relation 󵄨 󵄨 𝑓(|𝑥|) ∧ 𝑛𝑓(󵄨󵄨󵄨𝜑󵄨󵄨󵄨) ≤ 𝜆 𝑥 𝑓(𝑧) 𝑛 ∈ ℕ holds for any discrete functional 𝑓. Since 𝑓(𝜑) = 0, if 𝑓−1 (0) ∉ suppM (𝜑), only the case 𝑓−1 (0) ∈ suppM (𝜑) has to be considered, which will be done by establishing the inequality 󵄨 ̃ 󵄨󵄨 ̃ ∧ 𝑛 󵄨󵄨󵄨𝜑(𝑀) |𝑥(𝑀)| 󵄨󵄨 ≤ 𝜆 𝑥 for arbitrary 𝑀 ∈ suppM (𝜑), and 𝑛 ∈ ℕ, where 𝑥̃ and 𝜑̃ are the conditional representations (with respect to 𝑧) of 𝑥 and 𝜑. However, the related inequality holds, due to the fact that, by Proposition 6.8, the function 𝑥̃ is bounded on the compact set suppM (𝜑). This completes the proof of assertions (1) and (2) of the theorem. For further frequent use we introduce the set MΦ (𝐸) = 𝐺(Φ1 (𝐸)) =

⋃ 𝐺𝜑 = {𝑀 ∈ M(𝐸) : ∃𝜑 ∈ Φ1 (𝐸), 𝜑 ∉ 𝑀}, 𝜑∈Φ1 (𝐸)

sometimes abbreviated to MΦ . Much information on the finite elements of 𝐸 is related to topological properties of M(𝐸), and its subset MΦ (𝐸), where the latter will be equipped with the induced topology from (M(𝐸), 𝜏ℎ𝑘 ). The subspace MΦ (𝐸) has interesting properties, which makes its investigation very important and, in particuar, it is used for the characterization of totally finite and selfmajorizing elements in the vector lattice 𝐸. The sufficiency of Φ1 (𝐸)

̃ 3 𝑀 = 𝑓−1 (0) ∈ 𝐺𝜑 means 𝜑 ∉ 𝑀, thus 𝑓(𝜑) = 𝜑(𝑀) ≠ 0. 4 𝑓(𝑥) = 0 for any discrete functional 𝑓 implies 𝑥 = 0.

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means (see p. 106) that for each 𝑀 ∈ M there is a 𝜑 ∈ Φ1 (𝐸), such that 𝜑 ∉ 𝑀, i. e., 𝑀 ∈ 𝐺𝜑 and 𝑀 ∈ MΦ . It is clear that Φ1 (𝐸) is a sufficient set in the vector lattice 𝐸 if and only if MΦ = M. First of all, we are now able to complement the properties of the conditional representation. Theorem 6.11. Let 𝐸 be a vector lattice 𝐸 and 0 ≤ 𝑧 ∈ 𝐸. Then for each element 𝑥 ∈ 𝐸, the conditional representation 𝑥̃ (with respect to 𝑧) is continuous on the set 𝐺𝑧 ∩ MΦ . Proof. It may be assumed that 𝑥 ≥ 0. We put 𝐺 = 𝐺𝑧 ∩ MΦ , take any point 𝑀0 ∈ 𝐺, and prove the continuity of 𝑥̃ at the point 𝑀0 . This will be done after it is shown that the function 𝑥̃ coincides with the continuous function 𝑦̃ (for some 𝑦 ∈ 𝐼𝑧 ) in some neighborhood of the point 𝑀0 . In view of Theorem 6.10, we may fix a finite element 𝜑 ≥ 0 such that 𝑀0 ∈ 𝐺𝜑 and suppM (𝜑) ⊂ 𝐺. Then the element 𝑧 is one of the majorants of 𝜑 and, according to Proposition 6.9, the function 𝜑̃ is continuous on the set 𝐺. Without ̃ 0 ) > 𝑥(𝑀 ̃ 0 ). Otherwise, replace 𝜑 by the element loss of generality we may assume 𝜑(𝑀 ̃ ̃ 0 ) = 2𝜀 and 𝑦 = 𝑥 ∧ 𝜑. Then 𝜀 > 𝜆𝜑 for sufficiently large 𝜆 > 0. Let 𝜑(𝑀0 ) − 𝑥(𝑀 ̃ 0 ) = 𝑥(𝑀 ̃ 0 ) and the function 𝑦̃ is continuous on 𝐺. Therefore in some 0, 𝑦 ∈ 𝐼𝑧 , 𝑦(𝑀 (sufficiently small) neighborhood 𝑉 ⊂ 𝐺 of the point 𝑀0 the following inequalities ̃ ̃ 0 ) + 𝜀 = 𝑥(𝑀 ̃ 0 ) + 𝜀, 𝑦(𝑀) ≤ 𝑦(𝑀 ̃ ̃ 0 ) − 𝜀 = 𝑥(𝑀 ̃ 0 ) + 𝜀 for 𝑀 ∈ 𝑉 𝜑(𝑀) > 𝜑(𝑀 ̃ ̃ ̃ hold. Consequently, 𝑦(𝑀) = 𝑥(𝑀) ∧ 𝜑(𝑀), 𝑀 ∈ 𝑉. So the function 𝑥̃ coincides with the continuous function 𝑦̃ in the neighborhood 𝑉 of 𝑀0 . Together with Proposition 6.8 the theorem below is yielded. Theorem 6.12. Let 𝐸 be a vector lattice and 0 ≤ 𝑧 ∈ 𝐸. Then the function 𝑥̃ is continuous on the set 𝐺 = 𝐺𝑧 ∩ MΦ and bounded on each compact subset of 𝐺𝑧 . A topological Hausdorff space 𝑄 is called 𝜎-compact if it is the countable union of compact subsets. If 𝑄 is locally compact and 𝜎-compact topological space⁵ then a sequence (𝑄𝑘)𝑘∈ℕ of compact subsets of 𝑄 exists such that 𝑄 = ⋃𝑘∈ℕ 𝑄𝑘 and 𝑄𝑘 ⊂ int (𝑄𝑘+1 ) (see [31, Chap. I.9]). Theorem 6.10 has some interesting corollaries which are collected in the next theorem. Theorem 6.13. Let 𝐸 be a radical-free Archimedean vector lattice. Then (1) for 𝜑 ∈ Φ1 (𝐸) the subset 𝐺𝜑 ⊂ M(𝐸) is locally compact and 𝜎-compact; (2) the subspace MΦ (𝐸) is locally compact; (3) if the space M(𝐸) is compact then an order unit 𝐸 exists, and in particular 𝐸 = Φ1 (𝐸);

5 In [31] such locally compact spaces are called countable at infinity.

124 | 6 Topological characterization of finite elements (4) if for some point 𝑀0 ∈ M(𝐸) a compact neighborhood 𝑉exists, then a finite element 𝜑0 ∈ Φ1 (𝐸) exists, such that 𝑀0 ∈ 𝐺𝜑0 ⊂ 𝑉 (and 𝑀0 ∈ MΦ (𝐸)). Proof. (1) For the element 𝜑 and one of its 𝐸-majorants 𝑧, the assertion follows from ∞ 󵄨 ̃ 󵄨󵄨 1 ̃ ≠ 0} = ⋃ {𝑀 ∈ 𝐺𝑧 : 󵄨󵄨󵄨𝜑(𝑀) 𝐺𝜑 = {𝑀 ∈ 𝐺𝑧 : 𝜑(𝑀) 󵄨󵄨 ≥ }, 𝑛 𝑛=1

since 𝜑̃ is a continuous function on 𝐺𝑧 , and for any 𝛿 > 0 the set ̃ ≥ 𝛿} {𝑀 ∈ 𝐺𝑧 : |𝜑(𝑀)| is compact as a closed subset of the (compact) support of the finite element 𝜑. (2) follows from (1). (3) In view of Theorem 5.14 (3), the compactness of M(𝐸) implies the existence of an element 𝑧 ∈ 𝐸 such that M(𝐸) = 𝐺𝑧 . Then, on one hand, for each 𝑥 ∈ 𝐸 there is suppM (𝑥) ⊂ 𝐺𝑧 = M, which shows that 𝑧 is an 𝐸-majorant for the element 𝑥, and on the other hand, |𝑥| ∧ 𝑛 |𝑥| ≤ 𝑐𝑥𝑧 implies |𝑥| ≤ 𝑐𝑥 𝑧 for 𝑥 ∈ 𝐸, which is equivalent to 𝐸 = 𝐼𝑧 . It remains to refer to Theorem 3.6 in order to conclude 𝐸 = Φ(𝐸). (4) Since the system (𝐺𝑥 )𝑥∈𝐸 is a basis of the topology 𝜏ℎ𝑘 on M(𝐸), the open set int(𝑉) has a representation as int(𝑉) = ⋃ 𝐺𝑥 , where the union is extended over some collection of elements 𝑥 ∈ 𝐸 depending on 𝑉. Then an element 𝑥0 ∈ 𝐸 exists with 𝑀0 ∈ 𝐺𝑥0 ⊂ 𝑉. The compactness of 𝐺𝑥0 follows from 𝐺𝑥0 ⊂ 𝑉. Therefore 𝑥0 ∈ Φ1 (𝐸) and 𝑀0 ∈ MΦ (𝐸). Remark 6.14. The statements (2) and (4) of the theorem yield the two following facts: (1) the sets 𝐺𝜑 for 𝜑 ∈ Φ1 (𝐸) form the basis of the induced topology on MΦ (𝐸) (see statement (4) of the previous theorem); (2) MΦ (𝐸) is the largest open locally compact subspace in M(𝐸). In particular M(𝐸) is locally compact if and only if M(𝐸) = MΦ (𝐸); (3) Observe that in Example 5.17 (Kaplansky vector lattice), it is actually shown that MΦ (K) = [−2, 0) ∪ (0, 2], and that the topology 𝜏ℎ𝑘 coincides on MΦ (K) with the topology induced from the usual one in ℝ1 . For any point 𝑡 ∈ [−2, 2], 𝑡 ≠ 0 on some interval (𝑡 − 𝛿, 𝑡 + 𝛿) with sufficiently small 0 < 𝛿, such that 0 ∉ (𝑡 − 𝛿, 𝑡 + 𝛿), one can find a continuous function vanishing outside this interval. According to the description of the finite elements in K (see (b) on p. 23), such a function belongs to Φ1 (K) and so 𝑡 ∈ MΦ (K). For the point 𝑡 = 0 the neighborhoods are sets 𝐺𝑧 , where 𝑧(0) ≠ 0. Such a function cannot be a finite element in K and so 0 ∈ M(K) \ MΦ (K). (4) The space M(𝐸(𝑇), 𝜏ℎ𝑘 ) of the vector lattice 𝐸(𝑇) in Example 5.18, due to property (vii), is not locally compact, which is a further proof of the fact mentioned already in (2).

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6.1.3 Topological characterization of totally finite elements The totally finite elements in vector lattices are characterized now as follows. Theorem 6.15 (Topological characterization of totally finite elements). For a radicalfree Archimedean vector lattice 𝐸, the following assertions hold: (1) 𝜓 ∈ Φ2 (𝐸) ⇐⇒ suppM (𝜓) is compact and suppM (𝜓) ⊂ MΦ (𝐸); (2) Φ2 (𝐸) = Φ1 (Φ2 (𝐸)), i. e., each totally finite element has a totally finite majorant. Proof. (1) Let 𝜓 ∈ Φ2 (𝐸) and 𝑧 an 𝐸-majorant of 𝜓. Due to Theorem 6.10 suppM (𝜓) is a compact subset of 𝐺𝑧 . Since 𝑧 itself is a finite element, one has 𝐺𝑧 ⊂ MΦ , and so the provided condition is necessary. Inversely, if suppM (𝜓) is a compact subset of M = M(𝐸) which is contained in MΦ , then its open covering (𝐺𝑥)𝑥∈Φ1 (𝐸) contains a 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 finite subcovering, say {𝐺𝑥1 , 𝐺𝑥2 , . . . , 𝐺𝑥𝑁 }. The element 𝑦 = 󵄨󵄨󵄨𝑥1 󵄨󵄨󵄨 ∨ 󵄨󵄨󵄨𝑥2 󵄨󵄨󵄨 ∨ ⋅ ⋅ ⋅ ∨ 󵄨󵄨󵄨𝑥𝑁 󵄨󵄨󵄨 is finite, and 𝐺𝑦 ⊃ suppM (𝜓). According to statement (2) of Theorem 6.10, the element 𝑦 is a majorant of 𝜓. (2) Let 𝜓 ∈ Φ2 (𝐸). As has been proved so far, the set suppM (𝜓) is compact and contained in MΦ . We now construct a majorant for 𝜓 which lies in Φ2 (𝐸). If 𝑀 ∈ suppM (𝜓) by statement (2) of Theorem 6.13, there is a compact neighborhood 𝑉𝑀 of 𝑀 in M, such that in view of statement (4) of the aforementioned theorem there exists an element 𝜑𝑀 ∈ Φ1 (𝐸) with the property 𝑀 ∈ 𝐺𝜑𝑀 ⊂ 𝑉𝑀 . The open covering (𝐺𝜑𝑀 )𝑀∈suppM (𝜓) of suppM (𝜓) contains a finite subcovering {𝐺𝜑𝑀 , 𝐺𝜑𝑀 , . . . , 𝐺𝜑𝑀 } . 1

2

𝑘

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 For the element 𝑧 = 󵄨󵄨󵄨󵄨𝜑𝑀1 󵄨󵄨󵄨󵄨 ∨ 󵄨󵄨󵄨󵄨𝜑𝑀2 󵄨󵄨󵄨󵄨 ∨ ⋅ ⋅ ⋅ ∨ 󵄨󵄨󵄨󵄨𝜑𝑀𝑘 󵄨󵄨󵄨󵄨, we have the following inclusions: 𝑘

𝑘

suppM (𝜓) ⊂ 𝐺𝑧 = ⋃ 𝐺𝜑𝑀 ⊂ ⋃ 𝑉𝑀𝑖 , 𝑖

𝑖=1

from where

𝑖=1

𝑘

suppM (𝑧) = 𝐺𝑧 ⊂ ⋃ 𝑉𝑀𝑖 ⊂ MΦ 𝑖=1

follows. According to the first part of the proof 𝑧 ∈ Φ2 (𝐸), and, due to the second statement of Theorem 6.10, the element 𝑧 is a majorant of 𝜓. The second statement of the proved theorem shows that no further classification of finite elements is to be expected, i. e., the set of totally finite elements coincides with the set of totally finite elements possessing totally finite majorants (cf. with the situation for the finite elements, where, in general, Φ1 (Φ1 (𝐸)) = Φ2 (𝐸) ≠ Φ1 (𝐸). The corollaries below immediately follow from the first statement. Corollary 6.16. If MΦ (𝐸) is closed in the space M(𝐸), then Φ1(𝐸) = Φ2 (𝐸). In particular, one has Φ1 (𝐸) = Φ2 (𝐸) if MΦ (𝐸) = M(𝐸).

126 | 6 Topological characterization of finite elements Corollary 6.17. The system (𝐺𝜓 )𝜓∈Φ2(𝐸) forms a basis for the topology 𝜏ℎ𝑘 in MΦ (𝐸). In particular, MΦ (𝐸) = ⋃ 𝐺𝜓 . 𝜓∈Φ2 (𝐸)

Corollary 6.18. The canonical map 𝜔 is a homeomorphism from MΦ = 𝐺 (Φ1 (𝐸)) onto M (Φ2 (𝐸)). Proof. First of all, in view of the relations 𝐺(Φ2 (𝐸)) = 𝐺(Φ1 (𝐸)) = MΦ (𝐸),

(6.7)

the image of the set MΦ under the canonical map 𝜔 is a subset of M(Φ2 (𝐸)). Due to Proposition 6.3, where MΦ and 𝜔(MΦ ) were established as homeomorphic, it suffices to show only the surjectivity of the map 𝜔. The surjectivity of 𝜔 means that each maximal ideal in Φ2 (𝐸) can be extended to a maximal ideal in 𝐸. For this let 𝑁 be a maximal ideal in Φ2 (𝐸), i. e., 𝑁 ∈ M(Φ2 (𝐸)). Then there is an element 𝜑 ∈ Φ2 (𝐸) which does not belong to 𝑁. According to the second statement of the theorem there is a majorant of 𝜑 also belonging to Φ2 (𝐸). This allows us to extend 𝑁 to a maximal ideal 𝑀 in 𝐸 according to Proposition 6.4. Therefore, 𝜔(MΦ ) = M(Φ2 (𝐸)). Observe that the possibility provided by the last corollary, to extend maximal ideals in Φ2 (𝐸) to maximal ideals in 𝐸, is lost when Φ2 (𝐸) is replaced by Φ1 (𝐸), as Example 6.40 will show. The next corollaries follow immediately from (6.7) and statement (3) of Corollary 5.15. Corollary 6.19. The following conditions are equivalent: (1) the ideal Φ1 (𝐸) is complete in 𝐸; (2) the ideal Φ2 (𝐸) is complete in 𝐸; (3) MΦ is everywhere dense in M. Corollary 6.20. The ideal Φ2 (𝐸) is a vector lattice of type (Σ) if and only if MΦ (𝐸) considered with its induced topology is a 𝜎-compact noncompact space. Proof. We use the fact that MΦ (𝐸) is already locally compact. If the sequence 𝜓1 , 𝜓2 , . . . , 𝜓𝑛 , . . . satisfies the condition (Σ󸀠 ) in Φ2 (𝐸), then the set MΦ (𝐸) can be written as ∞

MΦ (𝐸) = ⋃ suppM (𝜓𝑛), 𝑛=1

and is therefore 𝜎-compact. Inversely, if MΦ is 𝜎-compact, i. e., MΦ = ⋃∞ 𝑛=1 𝐾𝑛 with the compact sets 𝐾𝑛 then, taking Corollary 6.17 into consideration, for each 𝑛 there is a finite number of elements 𝑘𝑛 𝜓1(𝑛) , 𝜓2(𝑛) , . . . , 𝜓𝑘(𝑛) in Φ2 (𝐸), such that 𝐾𝑛 ⊂ ⋃𝑖=1 𝐺𝜓𝑖(𝑛) . Put now 𝑛

𝜓(0) = 0

󵄨󵄨 󵄨󵄨 󵄨 󵄨 and 𝜓(𝑛) = 𝜓(𝑛−1) ∨ 󵄨󵄨󵄨󵄨𝜓1(𝑛) 󵄨󵄨󵄨󵄨 ∨ . . . ∨ 󵄨󵄨󵄨𝜓𝑘(𝑛) 󵄨󵄨󵄨 , 󵄨 𝑛󵄨

𝑛 = 1, 2, . . . .

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Then 𝐾𝑛 ⊂ 𝐺𝜓(𝑛) and, by the second statement of the theorem, the sequence (𝜓(𝑛) )𝑛∈ℕ satisfies the condition (Σ󸀠 ) in the vector lattice Φ2 (𝐸). Theorem 6.15 and its corollaries can be generalized to the case when the ideal Φ2 (𝐸) is replaced by an arbitrary ideal consisting of totally finite elements. Theorem 6.21. Let 𝐸0 be an ideal in the vector lattice 𝐸, 𝐸0 ⊂ Φ2 (𝐸), and 𝑥0 ∈ 𝐸. Then the following assertions hold: (1) 𝑥0 ∈ 𝐸0 if and only if the set suppM (𝑥0 ) is compact and contained in 𝐺(𝐸0 ), i. e., 𝐸0 = {𝑥 ∈ Φ1 (𝐸) : suppM (𝑥) ⊂ 𝐺(𝐸0 )}; (2) (3) (4) (5)

the system (𝐺𝑥 )𝑥∈𝐸0 is a basis of the topology in 𝐺(𝐸0 ); each element 𝜑 ∈ 𝐸0 has an 𝐸-majorant from 𝐸0 , in particular, 𝐸0 = Φ1 (𝐸0 ); the canonical map 𝜔 is a homeomorphism on 𝐺(𝐸0 ) onto M(𝐸0 ); 𝐸0 is a vector lattice of type (Σ) if and only if the set 𝐺(𝐸0 ) is 𝜎-compact and noncompact.

The proof of the theorem is analogous to that of Theorem 6.15 and its corollaries. An ideal consisting of totally finite elements can be generated by means of any open subset 𝐺 of MΦ . Then 𝐸𝐺 = {𝑥 ∈ Φ1 (𝐸) : suppM (𝑥) ⊂ 𝐺} is an ideal with 𝐸𝐺 ⊂ Φ2 (𝐸) and 𝐺(𝐸𝐺 ) = 𝐺. Indeed, it is easy to see that 𝐸𝐺 is an ideal. By means of the first statement of Theorem 6.15, the inclusion 𝐸𝐺 ⊂ Φ2 (𝐸) is clear, thus only 𝐺 = 𝐺(𝐸𝐺 ) = ⋃{𝐺𝑥 : 𝑥 ∈ 𝐸𝐺 } has to be proved, where 𝐺(𝐸𝐺 ) ⊂ 𝐺 is clear. So let 𝑀 ∈ 𝐺. Since MΦ is locally compact, there is a compact neighborhood 𝑉 of 𝑀 such that 𝑉 ⊂ 𝐺. Finally, statement (4) of Theorem 6.13 must be applied. It is interesting that the existence of nontrivial finite elements in a vector lattice 𝐸 also implies the existence of nontrivial totally finite elements. Theorem 6.22. Let 𝐸 be a vector lattice with Φ1 (𝐸) ≠ {0}. Then also Φ2 (𝐸) ≠ {0}. More precisely, for any 0 ≠ 𝜑 ∈ Φ1 (𝐸) a totally finite element 𝜓 exists, such that 𝜓 ≠ 0 and 𝜑 is a majorant for 𝜓. Proof. Let 𝜑 be an arbitrary nonzero finite element. Then 𝐺𝜑 ≠ 0. For 𝑀0 ∈ 𝐺𝜑 and a compact neighborhood 𝑉 ⊂ 𝐺𝜑 of 𝑀0 by Theorem 6.13 (4) an element 𝜓 ∈ Φ1 (𝑋) exists such that 𝑀0 ∈ 𝐺𝜓 ⊂ 𝐺𝜓 ⊂ 𝑉 ⊂ 𝐺𝜑 ⊂ MΦ . This shows that 𝜓 is totally finite, 𝜓 ≠ 0, and that 𝜑 is a majorant of 𝜓. Now we deal with conditions for the validity of the equality Φ1 (𝐸) = Φ2 (𝐸). A sufficient condition is expressed in terms of the topology in the space MΦ (see [131]).

128 | 6 Topological characterization of finite elements Definition 6.23. Let 𝑇 = (𝑇, 𝜏) be a topological space and 𝑇0 a subspace of 𝑇. Let {𝑂𝛼 }𝛼∈𝐼 be a basis of the topology in 𝑇0 induced by 𝜏. We say that {𝑂𝛼 }𝛼∈𝐼 has the 𝜎closure-property if the 𝜏-closure (in 𝑇) of the union of any countable number of basis sets is contained in 𝑇0 . If 𝑇0 is a closed subspace of 𝑇, this property obviously holds. Remark 6.24. The basis {𝐺𝜑 }𝜑∈Φ1 (𝐸) of MΦ has the 𝜎-closure-property if and only if the 𝜏ℎ𝑘 -closure of any locally compact, 𝜎-compact subset 𝐹 ⊂ M belongs to MΦ . Indeed, let 𝐹 ⊂ M be a locally compact, 𝜎-compact subset. Then, due to its 𝜎compactness, 𝐹 = ⋃∞ 𝑛=1 𝐾𝑛 , where 𝐾𝑛 is compact and 𝐾𝑛 ⊂ int(𝐾𝑛+1 ), 𝑛 = 1, 2, . . ., one has 𝐹 ⊂ MΦ . Since 𝐹 can also be represented as 𝐹 = ⋃∞ 𝑛=1 int(𝐾𝑛 ), it is open, and therefore, for some subset 𝐼 ⊂ Φ1 (𝐸) one has ∞

⋃ 𝐾𝑛 = 𝐹 = ⋃ 𝐺𝜑 . 𝑛=1

𝜑∈𝐼

Using the compactness of the sets 𝐾𝑛, for some countable subset 𝐼󸀠 ⊂ 𝐼 ⊂ Φ1 (𝑋), there is 𝐹 = ⋃𝜑∈𝐼󸀠 𝐺𝜑 . The 𝜎-closure-property of the basis {𝐺𝜑 }𝜑∈Φ1 (𝐸) now shows that 𝐹 ⊂ MΦ . Conversely, let 𝐺 = ⋃∞ 𝑖=1 𝐺𝜑𝑖 , for 0 < 𝜑1 , 𝜑2 , . . . , 𝜑𝑛 , . . . ∈ Φ1 (𝐸). Since any 𝐺𝜑𝑖 is locally compact and 𝜎-compact, the same property holds for 𝐺. In view of 𝐺 ⊂ MΦ , the assumption implies ∞

𝐺 = ⋃ 𝐺𝜑𝑖 ⊂ MΦ . 𝑖=1

Note that the necessity holds for any basis of MΦ which possesses the 𝜎-closureproperty. Theorem 6.25. If the subspace MΦ possesses a basis of the topology 𝜏ℎ𝑘 which has the 𝜎-closure-property, then Φ1 (𝐸) = Φ2 (𝐸). Proof. Let {𝑂𝛼 }𝛼∈𝐼 be a basis of the topology in MΦ which is induced by 𝜏ℎ𝑘 , and assume that this basis satisfies the condition of the theorem. Let 𝜑 ∈ Φ1 (𝐸) and 𝑧 ∈ 𝐸, 𝑧 ≥ 0 be a fixed 𝐸-majorant for 𝜑. Using the 𝜎-compactness of 𝐺𝜑 (see Theorem 6.13 (1)) for some subset 𝐼𝜑 ⊂ 𝐼 one has ∞

̃ 𝐺𝜑 = ⋃ {𝑀 ∈ 𝐺𝑧 : |𝜑(𝑀)| ≥ 𝑛=1

1 } = ⋃ 𝑂𝛼 , 𝑛 𝛼∈𝐼 𝜑

̃ ≥ thus for any 𝑛 = 1, 2, . . . the compact set {𝑀 ∈ 𝐺𝑧 : |𝜑(𝑀)| the system {𝑂𝛼 }𝛼∈𝐼𝜑 . Therefore, 𝐺𝜑 = ⋃ 𝑂𝛼󸀠 , 𝛼󸀠 ∈𝐼𝜑󸀠

1 } 𝑛

is openly covered by

6.1 Topological characterization of finite, totally finite and selfmajorizing elements

| 129

where 𝐼𝜑󸀠 is some countable set. By assumption, the closure 𝐺𝜑 = ⋃𝛼󸀠 ∈𝐼𝜑󸀠 𝑂𝛼󸀠 belongs to MΦ , i. e. 𝜑 ∈ Φ2 (𝐸). Remark 6.26. If a vector lattice 𝐸 satisfies the condition MΦ = MΦ , then each basis of the topology in MΦ has an even stronger property, namely the closure of any union of basis sets belongs to MΦ , i. e., the assumption of the theorem is satisfied and therefore Φ1 (𝐸) = Φ2 (𝐸) holds. Moreover, in the case of MΦ = MΦ for any 𝜑 ∈ Φ1 (𝐸), the closure of the set 𝐺𝜑 is obviously contained in MΦ . Due to the compactness of 𝐺𝜑 , the element 𝜑 is totally finite, which directly yields Φ1 (𝐸) = Φ2 (𝐸). We notice that the equality Φ1 (𝐸) = Φ2 (𝐸) is not nearly sufficient for the closedness of the subspace MΦ in M, as will be clear after Theorem 6.32 and Example 6.37.

6.1.4 Topological characterization of selfmajorizing elements The selfmajorizing elements of a vector lattice are characterized in the following theorem. Theorem 6.27 (Topological characterization of selfmajorizing elements). In a vector lattice 𝐸 the following assertion holds: a finite element 𝜑 is selfmajorizing if and only if suppM (𝜑) = 𝐺𝜑 . Proof. Remember that 𝐺|𝑥| = 𝐺𝑥 for each 𝑥 ∈ 𝐸; see p. 107. If 𝜑 is selfmajorizing , then |𝜑| is a majorant and the second statement of Theorem 6.10 implies 𝐺|𝜑| ⊆ 𝐺|𝜑|. Consequently, both sets coincide, and in view of the remark made earlier one has suppM (𝜑) = 𝐺𝜑 = 𝐺𝜑 . Conversely, if suppM (𝜑) = 𝐺𝜑 holds for a finite element 𝜑 ∈ 𝐸, and therefore also 𝐺𝜑 ⊆ 𝐺|𝜑| , then, again by the aforementioned theorem, 𝜑 is a selfmajorizing element. The next example shows the application of the proved theorem in order to find the selfmajorizing elements in the vector lattice c. Example 6.28. The selfmajorizing elements in the vector lattice c of all real convergent sequences. First we consider in detail the topology 𝜏ℎ𝑘 on the set M(c), and then use this information in order to find the selfmajorizing elements in c. It is well-known (see [8, Theorem 13.16]), that c under the supremum norm is an 𝐴𝑀space with unit 1 = (1, 1, . . .), and its norm dual c󸀠 is lattice isometric to ℝ ⊕ ℓ1 , where ∞ the isomorphism ℝ ⊕ ℓ1 󳨃→ c󸀠 is defined by 𝜆 lim 𝑥𝑛 + ∑𝑛=1 𝑎𝑛 𝑥𝑛 for 𝜆 ⊕ 𝑎 ∈ ℝ ⊕ ℓ1 , and 𝑛→∞

𝑥 = (𝑥𝑛)𝑛∈ℕ ∈ c. For each maximal ideal 𝑀 in c a discrete functional 𝑓 ∈ c󸀠 exists, such that 𝑓−1 (0) = 𝑀. The set of all discrete functionals on c consists of the positive multiples of the coordinate functionals 𝑥 󳨃→ 𝑥𝑖 for 𝑖 ∈ ℕ, and the multiples of the functional

130 | 6 Topological characterization of finite elements 𝑥 󳨃→ lim 𝑥𝑛. Hence, the maximal ideals in c are the sets 𝑛→∞

𝑀𝑖 = {𝑥 ∈ c : 𝑥𝑖 = 0} for 𝑖 ∈ ℕ

and c0 = {𝑥 ∈ c : lim 𝑥𝑛 = 0}. 𝑛→∞

Indeed, if 𝑀, 𝑀 ≠ c0 is a maximal ideal in c, and 0 ≠ (𝑥𝑛)𝑛∈ℕ = 𝑥 ∈ 𝑀, then lim 𝑥𝑛 ≠ 0. 𝑛→∞ Not all coordinates of 𝑥 can be different from zero. Otherwise 𝑥 would be an order unit and 𝑀 = c. So there is a nonempty subset 𝑁𝑥 ⊂ ℕ with 𝑥𝑛 = 0 for all 𝑛 ∈ 𝑁𝑥 . Since 𝑀 is maximal, the set 𝑁𝑥 consists of only one element, say 𝑖, the same for all 𝑥 ∈ 𝑀. Therefore 𝑀 = 𝑀𝑖 and M(c) = {𝑀𝑖 : 𝑖 ∈ ℕ} ∪ {c0}. We now describe the open and closed subsets in the space (M(c), 𝜏ℎ𝑘 ). Since c is radical-free, M(c) is a Hausdorff space by Theorem 5.16, and therefore the one-point sets {𝑀𝑖 } and {c0} are closed. We show that each set {𝑀𝑖 } (𝑖 ∈ ℕ) is also open. Indeed, let 𝑖 ∈ ℕ be a fixed index. By the definition of the topology 𝜏ℎ𝑘 , the set M(c) \ {𝑀𝑖 } is closed (and therefore the set {𝑀𝑖 } is open), if 𝑀𝑖 ⊉ ⋂ 𝑀𝑗 ∩ c0 .

(6.8)

𝑗=𝑖̸

This is indeed the case, because the right-hand side of (6.8) also contains all elements 𝑥 ∈ c0 , such that 𝑥𝑗 = 0 for all 𝑗 ≠ 𝑖, and 𝑥𝑖 ≠ 0, which are not elements of 𝑀𝑖 . In contrast, the one-point set {c0} is not open, as c0 ⊇ ⋂𝑖∈ℕ 𝑀𝑖 = {(0, 0, 0, . . .)} implies c0 ∈ M(c) \ {c0}. If a sequence 𝑥 ∈ c has infinitely many zero coordinates, then its limit is 0, and hence 𝑥 belongs to c0 . So c0 ⊇ ⋂𝑗∈N 𝑀𝑗 also holds if N is an arbitrary infinite subset of ℕ. Therefore, we conclude that none of these sets {𝑀𝑗 : 𝑗 ∈ N} is closed, as c0 is in the closure {𝑀𝑖 : 𝑖 ∈ N}, but not in the set itself. This implies that a set containing c0 can be open only if its complement is finite. Differently phrased, for each subset O ⊆ M(c) with c0 ∈ O, a number 𝑛0 ∈ ℕ must exist such that 𝑀𝑖 ∈ O for all 𝑖 ≥ 𝑛0 . Hence for a subset O ⊆ M(c) to be open there are two possibilities: (i) O is the complement of a finite set of points 𝑀𝑖 . Then O is also closed and c0 ∈ O; (ii) O = {𝑀𝑖 : 𝑖 ∈ N ⊆ ℕ}, i. e., a collection of points of kind 𝑀𝑖 . In this case, O is also closed if and only if N is a finite subset. On the subset {𝑀𝑖 : 𝑖 ∈ ℕ} = M(c) \ {c0} the topology 𝜏ℎ𝑘 induces the discrete one. Since the element 1 = (1, 1, 1, ...) is an order unit in c, according to Proposition 3.44 one has Φ1 (c) = Φ2 (c) = Φ3 (c) = c, and in particular, M(c) = MΦ (c). In order to identify the selfmajorizing elements in c, we apply Theorem 6.27. Hence our aim is to find those elements 𝑥, for which the set 𝐺𝑥 is both open and closed in M(c). Let 𝑥 = (𝑥𝑖 )𝑖∈ℕ ∈ c. Then 𝐺𝑥 = {𝑀 ∈ M(c) : 𝑥 ∉ 𝑀} is an open-closed set as in (i) if and only if lim 𝑥𝑖 ≠ 0, 𝑖→∞

whereas 𝐺𝑥 is open-closed as in (ii) if and only if 𝑥𝑖 ≠ 0 for at most finite many 𝑖 ∈ ℕ. According to Theorem 6.27 we get 𝑆(c) = {𝑥 ∈ c : lim 𝑥𝑖 ≠ 0} ∪ {𝑥 ∈ c : 𝑥𝑖 ≠ 0 for at most finite many 𝑖 ∈ ℕ}. 𝑖→∞

Therefore, e. g., among the sequences 𝑥 = (1, 4, 9, . . . , 𝑛2 , 0, 0, . . .), 𝑦 = (2, 32 , . . . , 1+ 1𝑛 , . . .), and 𝑧 = (1, 12 , 13 , . . . , 1𝑛 , . . .), the elements 𝑥 and 𝑦 are selfmajorizing elements, but 𝑧 is not.

6.2 Relations between the ideals of finite, totally finite and selfmajorizing elements

|

131

Due to the relation c = Φ3 (c) = 𝑆+ (c) − 𝑆+ (c) = c, each element of c allows a representation as a difference of two positive selfmajorizing elements. The above element 𝑧 has, for example, the representation 𝑧 = (𝑧 + 𝜀1) − 𝜀1 for every 𝜀 > 0.

6.2 Relations between the ideals of finite, totally finite and selfmajorizing elements After the topological characterization of all three types of finite elements we are now able to study the relations between all kinds of finite elements in a given vector lattice more substantially. Notice that in [131], such an investigation has been undertaken without attracting the selfmajorizing elements. In order to get the whole picture, i. e., all possible inclusions between the ideals Φ2 (𝐸), Φ1 (𝐸), and the whole (nontrivial) vector lattice 𝐸, we first provide the tableau of [131], which exactly shows the possible and impossible relations between the vector lattice 𝐸 and the ideals Φ1 (𝐸), Φ2 (𝐸) (Table 6.1). Later in this section we complete Tableau 6.1 by including the ideal Φ3 (𝐸). Table 6.1. The relations between Φ𝑖 (𝐸) for 𝑖 = 1, 2 Case a) b) c) d) e) f) g)

{0}

⊆ ≠ = ≠ ≠ ≠ = =

Φ2 (𝐸)

⊆ ≠ = = = ≠ ≠ ≠

Φ1 (𝐸)

⊆ ≠ ≠ ≠ = = = ≠

𝐸

Possible yes yes yes yes no no no

Among them, only cases a)–d) are admissible, i. e., cases e), f), and g) are inconsistent. Examples of cases a) and b) are the Kaplansky-type vector lattice discussed in Examples 3.4 and 3.5 in Section 3.1. Examples of case c), with {0} ≠ Φ2 (𝑋) = Φ1 (𝑋) ≠ 𝑋, are the vector lattice M (𝑇) of Radon measures on a locally compact 𝜎-compact space [129] and several vector lattices consisting of infinite matrices [91]; see e. g., Examples 6.35 and 6.37. An example of case d), with {0} ≠ Φ2 (𝐸) = 𝐸, provides the vector lattice K((0, +∞)) of all continuous functions on (0, +∞) having compact support. If a vector lattice 𝐸 satisfies the condition Φ1 (𝐸) = 𝐸, then obviously Φ2 (𝐸) = Φ1 (𝐸) (see p. 26). As a consequence, cases e) and f) are inconsistent. In Theorem 6.22 it was proved, in particular, that the presence of nontrivial finite elements implies the existence of nontrivial totally finite elements as well. Thus, case g) is also inconsistent.

132 | 6 Topological characterization of finite elements The starting points for incorporating selfmajorizing elements, more exactly the ideal Φ3 (𝐸), into our investigation are the relations: I {0} = Φ2 (𝐸) = Φ1 (𝐸) ⊊ 𝐸 II {0} ⊊ Φ2 (𝐸) ⊊ Φ1 (𝐸) ⊊ 𝐸 III {0} ⊊ Φ2 (𝐸) = Φ1 (𝐸) ⊊ 𝐸 IV {0} ⊊ Φ2 (𝐸) = Φ1 (𝐸) = 𝐸. In order to include the ideal Φ3 (𝐸), we only have to deal with cases II, III, and IV. We show that there are vector lattices 𝐸 satisfying one of the relations II, III, or IV, such that Φ3 (𝐸) = {0}. This means the following relations are possible: II= {0} = Φ3 (𝐸) ⊊ Φ2 (𝐸) ⊊ Φ1 (𝐸) ⊊ 𝐸 III= {0} = Φ3 (𝐸) ⊊ Φ2 (𝐸) = Φ1 (𝐸) ⊊ 𝐸 IV= {0} = Φ3 (𝐸) ⊊ Φ2 (𝐸) = Φ1 (𝐸) = 𝐸. Examples of III= and IV= are the vector lattices 𝐶(ℝ) and K(ℝ) respectively. The Kaplansky vector lattice K (see Example 3.4) provides an example of case II= . To see this, we first show that inf 𝑡∈𝐼 𝜑(𝑡) > 0 if 𝜑 does not vanish on 𝐼, whenever 𝜑 is a selfmajorizing element, and 𝐼 an arbitrary interval in 𝑇 = [−2, 2] \ {1, 12 , 13 , . . . , 1𝑛 , . . .}. Indeed, assume that on some interval 𝐼 ⊂ 𝑇 the infimum of 𝜑 on 𝐼 is 0. Then there is a sequence (𝑡𝑛)𝑛∈ℕ ⊆ 𝐼 with 𝜑(𝑡𝑛) > 0 and 𝜑(𝑡𝑛) → 0. For fixed 𝑛, choose 𝑚 ∈ ℕ sufficiently large such that 𝑚𝜑(𝑡𝑛) ≥ 1. Then for the function 1 ∈ K one has 1 = (1 ∧ 𝑚𝜑)(𝑡𝑛) ≤ 𝑐1 𝜑(𝑡𝑛), and hence 𝜑(𝑡𝑛) ≥ 𝑐1 for any 𝑛 ∈ ℕ. This however contradicts the relation 𝜑(𝑡𝑛) → 0. 1 Now we are able to show that a positive selfmajorizing element 𝜑 can only be the zero element in K. We need the functions 𝑒𝜈 ∈ K for 𝜈 = 1, 2, . . . defined as 𝜈 1 𝑒𝜈 (𝑡) = ∑𝑘=1 |𝑘𝑡−1| ; see (3.3). Assume the contrary, i. e., let 𝜑 be a positive selfmajorizing element for which a point 𝑠0 ∈ 𝑇 exists such that 𝜑(𝑠0 ) > 0. If 𝑠0 ∈ [−2, 0], then inf 𝑡∈[−2,0] 𝜑(𝑡) > 0, which contradicts⁶ 𝜑 ∈ Φ2 (𝐸). If 𝑠0 ∈ (1, 2], then for the function 𝑒1 ∈ K there is a constant 𝑐𝑒1 > 0 with 𝑒1 ∧ 𝑚𝜑 ≤ 𝑐𝑒1 𝜑 for all 𝑚 ∈ ℕ. Due to inf 𝑡∈(1,2] 𝜑(𝑡) > 0, the equality 𝑒1 (𝑡) ∧ 𝑚𝜑(𝑡) = 𝑒1 (𝑡) holds for all 𝑡 ∈ (1, 2] if 𝑚 is sufficiently large. Hence, (6.9)

𝑒1 (𝑡) ≤ 𝑐𝑒1 𝜑(𝑡) on the interval (1, 2]. Since lim |𝑡 − 1|𝑒1 (𝑡) 𝑡→1

>

0, the inequality (6.9) implies

lim |𝑡 − 1| 𝜑 (𝑡) > 0. This prevents 𝜑 from vanishing identically on the interval ( 12 , 1). 𝑡→1

In particular, there is a point 𝑠1 ∈ ( 12 , 1), such that 𝜑(𝑠1 ) > 0. Analogously, by means of the function 𝑒2 , the existence of a point 𝑠2 ∈ ( 31 , 12 ) with 𝜑(𝑠2 ) > 0 can be shown.

6 See the description of totally finite elements of K, i. e., (c) on p. 24.

6.2 Relations between the ideals of finite, totally finite and selfmajorizing elements

| 133

So by using the functions 𝑒𝑛 inductively, a sequence (𝑠𝑛)𝑛∈ℕ is constructed such that 1 𝑠𝑛 ∈ ( 𝑛+1 , 1𝑛 ), 𝜑(𝑠𝑛 ) > 0 for each 𝑛 ∈ ℕ and 𝑠𝑛 → 0. Again, we get a contradiction to the fact 𝜑 ∈ Φ2 (K). Thus, 𝑆+(K) = {0} and therefore Φ3 (K) = {0}. 1 If 𝑠0 ∈ ( 𝑛+1 , 𝑛1 ], then we proceed in a similar manner as before. Next we show that for a vector lattice 𝐸 the following inclusions are also possible: II≠ {0} ≠ Φ3 (𝐸) ⊊ Φ2 (𝐸) ⊊ Φ1 (𝐸) ⊊ 𝐸 III≠ {0} ≠ Φ3 (𝐸) ⊊ Φ2 (𝐸) = Φ1 (𝐸) ⊊ 𝐸 IV≠ {0} ≠ Φ3 (𝐸) ⊊ Φ2 (𝐸) = Φ1 (𝐸) = 𝐸. Consider the vector lattice 𝐸 defined as 𝐸 = 𝐹 ⊕ 𝐶[0, 1], where 𝐹 is an Archimedean vector lattice, and 𝐶[0, 1] the vector lattice of all continuous functions on the interval [0, 1]. The order in 𝐸 is defined coordinatewise, i. e., for 𝑥, 𝑦 ∈ 𝐸, where 𝑥 = (𝑥1 , 𝑥2 ) and 𝑦 = (𝑦1 , 𝑦2 ), we have 𝑥 ≤ 𝑦 ⇔ 𝑥1 ≤ 𝑦1 and 𝑥2 ≤ 𝑦2 . Then 𝐸 is an Archimedean vector lattice, and for the elements 𝑥 = (𝑥1 , 𝑥2 ), 𝑦 = (𝑦1 , 𝑦2 ), 𝑧 = (𝑧1 , 𝑧2 ) of 𝐸, and numbers 𝑐 > 0, 𝑛 ∈ ℕ one has |𝑦| ∧ 𝑛|𝑥| ≤ 𝑐𝑧

⇐⇒

|𝑦1 | ∧ 𝑛|𝑥1 | ≤ 𝑐𝑧1 in 𝐹 and |𝑦2 | ∧ 𝑛|𝑥2 | ≤ 𝑐𝑧2 in 𝐶 [0, 1] .

Therefore, an element 𝑥 = (𝑥1 , 𝑥2 ) ∈ 𝐸 is finite (totally finite, selfmajorizing) if and only if both 𝑥1 is finite in 𝐹, and 𝑥2 is finite in 𝐶[0, 1] (totally finite, selfmajorizing, respectively). The element 𝜓 = (𝜓1 , 𝜓2 ), with 𝜓1 = 0, and 𝜓2 = 1 is a (nontrivial) positive selfmajorizing element in 𝐸, since for 𝑥 = (𝑥1 , 𝑥2 ) ∈ 𝐸 one has |𝑥2 | ∧ 𝑛1) ≤ 𝑐2 (0, 1) = 𝑐2 𝜓 |𝑥| ∧ 𝑛𝜓 = (|𝑥1 | ∧ 𝑛𝜓1 , |𝑥2 | ∧ 𝑛𝜓2 ) = (0, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ≤𝑐2 1

for some 𝑐2 > 0. Therefore, {(0, 0)} ≠ Φ3 (𝐸). Because of Φ1 (𝐶[0, 1]) = Φ2 (𝐶[0, 1]) = 𝐶[0, 1] the (total) finiteness of 𝑥1 in 𝐹 is responsible for the (total) finiteness of the element (𝑥1 , 𝑥2 ) in 𝐸. The consequence is that if any of the inclusions II, III, or IV holds in 𝐹, then it also holds in 𝐸. Further, the assumption⁷ Φ3 (𝐹) = {0} yields nontrivial totally finite elements which are not in Φ3 (𝐹). If 𝜑 ∈ 𝐹 is such an element, then (𝜑, 0) is a nonzero totally finite element in 𝐸 as well, however (𝜑, 0) ∉ Φ3 (𝐸). Hence Φ3 (𝐸) ⊊ Φ2 (𝐸). This shows that the inclusions II≠ , III≠ , and IV≠ are possible. In Proposition 3.44 we saw that the ideals Φ3 (𝐸), Φ2 (𝐸), and Φ1 (𝐸) coincide with 𝐸 if the vector lattice 𝐸 possesses an order unit. In the vector lattice s, the inclusions III hold as well as Φ3 (s) = Φ2 (s); see Example 3.48. Consequently, vector lattices exist which satisfy the inclusions V≠ {0} ⊊ Φ3 (𝐸) = Φ2 (𝐸) = Φ1 (𝐸) ⊊ 𝐸 VI≠ {0} ⊊ Φ3 (𝐸) = Φ2 (𝐸) = Φ1 (𝐸) = 𝐸.

7 E. g., 𝐹 = K.

134 | 6 Topological characterization of finite elements It remains open whether a vector lattice 𝐸 exists which satisfies the inclusion {0} ⊊ Φ3 (𝐸) = Φ2 (𝐸) ⊊ Φ1 (𝐸) ⊊ 𝐸. Table 6.2 shows all currently known possible relations between the ideals of selfmajorizing, totally finite and finite elements in a vector lattice. The summary also contains the corresponding examples. Table 6.2. The relations between Φ𝑖 (𝐸) for 𝑖 = 1, 2, 3 Relations =

II III= IV= II≠ III≠ IV≠ V≠ VI≠

{0} {0} {0} {0} {0} {0} {0} {0}

= Φ3 (𝐸) = Φ3 (𝐸) = Φ3 (𝐸) ⊊ Φ3 (𝐸) ⊊ Φ3 (𝐸) ⊊ Φ3 (𝐸) ⊊ Φ3 (𝐸) ⊊ Φ3 (𝐸)

Example ⊊ Φ2 (𝐸) ⊊ Φ2 (𝐸) ⊊ Φ2 (𝐸) ⊊ Φ2 (𝐸) ⊊ Φ2 (𝐸) ⊊ Φ2 (𝐸) = Φ2 (𝐸) = Φ2 (𝐸)

⊊ Φ1 (𝐸) ⊊ = Φ1 (𝐸) ⊊ = Φ1 (𝐸) = ⊊ Φ1 (𝐸) ⊊ = Φ1 (𝐸) ⊊ = Φ1 (𝐸) = = Φ1 (𝐸) ⊊ = Φ1 (𝐸) =

𝐸 𝐸 𝐸 𝐸 𝐸 𝐸 𝐸 𝐸

K 𝐶(ℝ) K(ℝ) K ⊕ 𝐶[0, 1] 𝐶(ℝ) ⊕ 𝐶[0, 1] K(ℝ) ⊕ 𝐶[0, 1] s 𝐶[0, 1]

6.3 The topological space M for vector lattices of type (Σ) The importance of the subspace MΦ has already been mentioned. Especially for Archimedean vector lattices of type (Σ), quite a lot is known about MΦ . In particular, for the existence of representations with the desirable property that a continuous function with a compact support is assigned to any finite element, the closedness and 𝜎compactness of MΦ in M is decisive; see Chapter 9. Remember that a vector lattice 𝐸 is of type (Σ) if it contains a sequence of elements 󸀠 (𝑒𝑛)∞ 𝑛=1 with the property (Σ ) and does not possess order units. It has already been mentioned that a vector lattice of type (Σ) is radical-free. In that case, the closedness of MΦ (𝐸) in M(𝐸) implies its 𝜎-compactness, and by Theorem 6.15 the equality Φ2 (𝐸) = Φ1 (𝐸). We start with the following theorem. Theorem 6.29. Let 𝐸 be an Archimedean vector lattice of type (Σ) and 𝐹 a closed subset of M(𝐸). If 𝐹 is in MΦ (𝐸), then 𝐹 is 𝜎-compact. 󸀠 Proof. For a sequence (𝑒𝑛)∞ 𝑛=1 which satisfies the condition (Σ ) in 𝐸, we consider in MΦ = MΦ (𝐸) the subsets

𝑆𝑛 = {𝑀 ∈ MΦ : 𝑒𝑛 ∉ 𝑀},

i. e., 𝑆𝑛 = 𝐺𝑒𝑛 ∩ MΦ ,

and have ⋃∞ 𝑛=1 𝑆𝑛 = MΦ . If it is assumed that 𝐹 ⊂ MΦ is not 𝜎-compact, then there is a natural number 𝑛, such that 𝐹 ∩ 𝑆𝑛 cannot be contained in any countable union

6.3 The topological space M for vector lattices of type (Σ)

|

135

of compact sets. Without loss of generality we may assume that the set 𝐴 1 = 𝐹 ∩ 𝑆1 has this property. Denote by S the system of all those subsets of 𝐴 1 which are contained in some 𝜎-compact set of the space M(𝐸). Further on, let 𝑥̃ be the conditional representation of 𝑥 ∈ 𝐸 with respect to 𝑒1 . According to Theorem 6.12, all conditional representations are continuous on 𝑆1 . Since ∞

𝐴 1 = ⋃ {𝑀 ∈ 𝐴 1 : 𝑒̃2 (𝑀) ≤ 𝑚}, 𝑚=1

there is a natural number 𝑛2 , such that the set 𝐴 2 = {𝑀 ∈ 𝐴 1 : 𝑒̃2 (𝑀) ≤ 𝑛2 } does not belong to the system S. The set 𝐴 2 has the representation ∞

∞

𝐴 2 = ⋃ {𝑀 ∈ 𝐴 2 : 𝑒̃3 (𝑀) ≤ 𝑚} = ⋃ {𝑀 ∈ 𝐴 1 : 𝑒̃2 (𝑀) ≤ 𝑛2 , 𝑒̃3 (𝑀) ≤ 𝑚}, 𝑚=1

𝑚=1

which yields the existence of a number 𝑛3 , such that the set 𝐴 3 = {𝑀 ∈ 𝐴 1 : 𝑒̃2 (𝑀) ≤ 𝑛2 , 𝑒̃3 (𝑀) ≤ 𝑛3 } does not belong to S. By induction we produce the sets 𝐴 𝑘 = {𝑀 ∈ 𝐴 1 : 𝑒̃2 (𝑀) ≤ 𝑛2 , 𝑒̃3 (𝑀) ≤ 𝑛3 , . . . , 𝑒̃𝑘 (𝑀) ≤ 𝑛𝑘 }, where the number 𝑛𝑘 is chosen such that 𝐴 𝑘 ∉ S, 𝑘 = 2, 3, . . .. In the set 𝐴 1 we consider the ultrafilter U, which contains the sets 𝐴 𝑘 , 𝑘 = 2, 3, . . . and the complementary sets (with respect to 𝐴 1 ) from all sets of the system S. By construction of the sets 𝐴 𝑘, each of the functions 𝑥̃ for 𝑥 ∈ 𝐸 is bounded on at least one of them⁸, and therefore possesses ̃ the (finite) limit lim 𝑥(𝑀) = 𝑓(𝑥), with respect to the ultrafilter U. For 𝜑 ∈ Φ1 (𝐸), the U

set 𝐺𝜑 ∩ 𝐴 1 is a subset even of the compact set suppM (𝜑), and therefore belongs to the system S. As a consequence one has 𝑓(𝜑) = 0

for any

𝜑 ∈ Φ1 (𝐸).

(6.10)

Due to 𝑓(𝑒1 ) = 1, the functional 𝑓 is nontrivial and discrete. In order to obtain a contradiction, we construct a finite element on which 𝑓 does not vanish. The maximal ̃ 0 ) = 𝑓(𝑥). The relation ideal 𝑀0 = 𝑓−1 (0) belongs to 𝑆1 , and one has 𝑥(𝑀 ̃ 0 ) = 𝑓(𝑥) ̃ 𝑥(𝑀) 󳨀→ 𝑥(𝑀 U

𝑥∈𝐸

implies the convergence U 󳨀→ 𝑀0 , with respect to the topology 𝜏ℎ𝑘 (cf. Theorem 5.14 (4)). Therefore, in view of the closedness of 𝐹, one has 𝑀0 ∈ 𝐹. Consequently, 𝑀0 ∈ MΦ and due to Theorem 6.13 (4), there is a finite element 𝜑0 , such that 𝑀0 ∈ 𝐺𝜑0 . This means 𝑓(𝜑0 ) ≠ 0, a contradiction of (6.10).

8 This holds, since for each 𝑥 ∈ 𝐸 there are numbers 𝑛 and 𝜆𝑥 , such that |𝑥| ≤ 𝜆 𝑥 𝑒𝑛 .

136 | 6 Topological characterization of finite elements This leads immediately to the following corollary. Corollary 6.30. Let 𝐸 be a vector lattice of type (Σ). If the set MΦ (𝐸) is closed in M(𝐸), then the space MΦ (𝐸) is 𝜎-compact and, according to Corollary 6.16, Φ2 (𝐸) = Φ1 (𝐸) holds. The inverse assertion, in general, is not true, i. e., Φ2 (𝐸) = Φ1 (𝐸) does not imply the closedness of MΦ (𝐸); see Example 6.37. Remark 6.31. In particular, the following is true: if the space M(𝐸) = M of a vector lattice of type (Σ) is locally compact, i. e., one has M = MΦ , then M is 𝜎-compact. If MΦ ≠ MΦ , then MΦ need not be 𝜎-compact; see Example 6.35. For vector lattices 𝐸 of type (Σ), the important property of closedness of the set MΦ (𝐸) in M(𝐸) is characterized in the following theorem, where for the proof of its sufficiency we use uniform completeness (i. e. (𝑟)-completeness) of the vector lattice, i. e., every uniformly Cauchy sequence is relatively uniformly convergent (see p. 6). Theorem 6.32. Let 𝐸 be of type (Σ). For the closedness of MΦ(𝐸) in M(𝐸), it is necessary, and in case of the uniform completeness of 𝐸 also sufficient, that both of the following conditions hold: (i) Φ2 (𝐸) is a vector lattice of type (Σ); and (ii) Φ1 (𝐸) = Φ2 (𝐸). Proof. Let MΦ = MΦ . Then condition (ii) holds, due to Corollary 6.16. The necessity of condition (i) follows from Corollaries 6.30 and 6.20. In order to prove the sufficiency of the conditions, we assume the contrary: let the set MΦ not be closed. Take 𝑀0 ∈ MΦ \ ∞ 󸀠 MΦ and a sequence (𝑒𝜈 )∞ 𝜈=1 in 𝐸 which satisfies the condition (Σ ). Due to M = ⋃𝜈=1 𝐺𝑒𝜈 , one has 𝑀0 ∈ 𝐺𝑒𝑛 for some number 𝑛, where without loss of generality we assume 𝑛 = 1. Then 𝑀0 ∈ 𝐺, where 𝐺 = 𝐺𝑒1 ∩ MΦ . The conditional representation 𝑥,̃ with respect to 𝑒1 of all 𝑥 ∈ 𝐸, is continuous on 𝐺 according to Theorem 6.11. Consider an ultrafilter U on 𝐺 which converges to 𝑀0 . For each positive element 𝑥 ∈ 𝐸 the relation ̃ ∧ 𝐶𝑒1(𝑀) = sup lim 𝑥̃ lim 𝑥(𝑀) U

𝐶>0 U

∧ 𝐶𝑒1 on 𝐺𝑒1 (see can be easily established. By using the continuity of the functions 𝑒𝑛̃ Proposition 6.9) one has lim 𝑒̃𝑛(𝑀) = sup lim 𝑒𝑛̃ ∧ 𝐶𝑒1 (𝑀) = sup 𝑒𝑛̃ ∧ 𝐶𝑒1 (𝑀0 ) ≤ 𝑒̃𝑛(𝑀0 ), U

𝐶>0 U

𝐶>0

i. e., lim 𝑒̃𝑛(𝑀) ≤ 𝑒̃𝑛(𝑀0 ), U

𝑛 = 1, 2, . . . .

(6.11)

According to Corollary 6.20, the assumption that Φ2 (𝐸) is a vector lattice of type (Σ) is equivalent to the 𝜎-compactness of the subspace MΦ . Let MΦ = ⋃∞ 𝑛=1 𝐾𝑛 , where 𝐾𝑛 are compact sets such that 𝐾𝑛 ⊆ int (𝐾𝑛+1 ) for all 𝑛 = 1, 2, . . .. Notice that any compact

6.3 The topological space M for vector lattices of type (Σ)

| 137

subset of MΦ is contained in one of the sets 𝐾𝑛 . Due to the inequalities (6.11), and the continuity of the functions 𝑒𝑛̃ on 𝐺, there are open sets 𝐺𝑛 ⊆ 𝐺 with the properties 𝐺𝑛 ∈ U,

𝐺𝑛 ⊇ 𝐺𝑛+1 ,

𝐺𝑛 ∩ 𝐾𝑛 = 0

𝑒̃𝑛(𝑀) < 2̃ 𝑒𝑛(𝑀0 ),

if 𝑀 ∈ 𝐺𝑛,

𝑛 = 1, 2, . . . .

(6.12)

Let points 𝑀𝑛 ∈ 𝐺𝑛 now be fixed. In view of the local compactness of MΦ , and statement (4) in Theorem 6.13, for any 𝑛 one finds a finite element 𝜑𝑛 ≥ 0, such that 𝑀𝑛 ∈ 𝐺𝜑𝑛 ⊂ 𝐺𝜑𝑛 ⊆ 𝐺𝑛.

(6.13)

̃𝑛 (𝑀) and 𝜆 𝑛 = 𝛼 1𝑛2 . The relations (6.13) imply 𝜑𝑛 ≤ 𝛼𝑛𝑒1 for Denote 𝛼𝑛 = max𝑀∈𝐺𝑛 𝜑 𝑛 𝑛 = 1, 2, . . ., and so the (𝑟)-completeness of the vector lattice 𝐸 guarantees the existence of the sum of the series ∑∞ 𝑛=1 𝜆 𝑛 𝜑𝑛 = 𝜓 in 𝐸. The element 𝜓 is not totally finite, since the set ∞

suppM (𝜓) ⊃ ⋃ suppM (𝜑𝑛 ) 𝑛=1

is not contained in any of the sets 𝐾𝑛 . We come to a contradiction of condition (ii) of the theorem if we show that 𝜓 is a finite element of 𝐸. For the latter we check whether for each 𝑛 there is a number 𝑐𝑛, such that for each 𝐶 > 0 the inequality 𝑒𝑛 ∧ 𝐶 𝜓 ≤ 𝑐𝑛𝑒1 holds. Since the set of all discrete functionals is total on 𝐸, it suffices to show that 𝑓(𝑒𝑛 ∧ 𝐶 𝜓) ≤ 𝑐𝑛 𝑓(𝑒1 ) for all 𝐶 > 0.

(6.14)

If 𝑀 = 𝑓−1 (0) ∉ 𝐺𝜓, i. e., 𝑓(𝜓) = 0, then (6.14) trivially holds. If 𝑀 ∈ 𝐺𝜓 ⊂ 𝐺𝑒1 , then instead of (6.14) the inequality ̃ 𝑒̃𝑛(𝑀) ∧ 𝐶 𝜓(𝑀) ≤ 𝑐𝑛 ,

𝐶>0

̃ will be established. In view of 𝜓(𝑀) > 0 for 𝑀 ∈ 𝐺𝜓 , this means that the boundedness of the function 𝑒𝑛̃ on the set 𝐺𝜓 has to be shown. It is clear that 𝑛−1

𝐺𝜓 ⊂ ( ⋃ suppM (𝜑𝑘 )) ∪ 𝐺𝑛 𝑘=1

holds. According to Theorem 6.11, the function 𝑒̃𝑛 is continuous on the set 𝐺 = 𝐺𝑒1 ∩ 𝑛−1 MΦ , and therefore bounded on the compact subset ⋃𝑘=1 suppM (𝜑𝑘 ). The function 𝑒̃𝑛 is bounded on 𝐺𝑛, due to (6.12). So the inequality 6.14 is shown, which completes the proof. Corollary 6.33. If, for a uniformly complete vector lattice 𝐸 of type (Σ), the conditions (i) and (ii) of the theorem hold, and if the set Φ1 (𝐸) is complete in 𝐸, then the space M is locally compact and 𝜎-compact.

138 | 6 Topological characterization of finite elements Indeed, in view of the theorem, one has MΦ = MΦ . The completeness of Φ1 (𝐸) in 𝐸 implies MΦ = M by means of Corollary 6.19, and so MΦ = M. The properties of M now follow immediately from the second Remark in 6.14 and Theorem 6.29. Even for a vector lattice 𝐸 in which the set MΦ is everywhere dense in M (or in other words, in which the set of finite elements is complete), the sufficiency of the last theorem may not hold if 𝐸 is not of type (Σ), fails to be uniformly complete, or the conditions (i) and (ii) are violated. This will be demonstrated by counterexamples in the next section.

6.4 Examples For a radical-free vector lattice 𝐸 consider now the following conditions: (a) 𝐸 is a vector lattice of type (Σ); (b) 𝐸 is a uniformly complete vector lattice; (i) Φ2 (𝐸) is vector lattice of type (Σ), i. e., MΦ (𝐸) is 𝜎-compact; (ii) Φ1 (𝐸) = Φ2 (𝐸). All these four conditions are used in Theorem 6.32 for the proof of the closedness of MΦ in M. They are essential and non of them can be dropped, as our next examples will demonstrate. Although the four Examples (6.34–6.39) are related to that theorem, each one is interesting enough on its own. We show that the proof of sufficiency in Theorem 6.32 fails if only one of the conditions is not satisfied. Observe that in each of these counterexamples the ideal Φ1 (𝐸) is complete, i. e., according to Corollary 5.15 (3), the equality MΦ = M holds. Therefore, the conclusion that MΦ is not closed will then follow if the inequality MΦ ≠ M is established. In order to check conditions (i) and (ii) in the corresponding vector lattices we will preliminarily describe the finite elements. Example 6.34. Example of an (r)-complete vector lattice 𝐸 not of type (Σ) with Φ1 (𝐸) = Φ2 (𝐸), where Φ2 (𝐸) is a vector lattice of type (Σ). Moreover, MΦ = M but MΦ ≠ M. At the same time this is a counterexample to Theorem 6.32, which shows that condition (a) (𝐸 is of type (Σ)) is not dispensable for MΦ to be closed in M. Let 𝐷 be the open unique disc in the plane, i. e., 𝐷 = {(𝑠, 𝑡) ∈ ℝ2 : 𝑠2 + 𝑡2 < 1}. Denote by 𝐸 the vector lattice of all continuous functions 𝑥 on 𝐷 for which the finite limit 𝐹(𝑥) = lim 𝑥(𝑠, 0) 𝑠→1−0

exists. It is easy to see that this vector lattice is uniformly complete. We show that the set of all finite elements in 𝐸 coincides with the set of all functions possessing compact support in 𝐷. Clearly, each function with a compact support in 𝐷 is a finite, obviously even a totally finite element of 𝐸. Conversely, let 0 < 𝜑 ∈ Φ1 (𝐸) and 𝑧 be one of its majo-

6.4 Examples

| 139

rants. If there would exist a sequence ((𝑠𝑛, 𝑡𝑛))𝑛∈ℕ in 𝐷 with 𝑠2𝑛 + 𝑡2𝑛 󳨀→ 1 and 𝜑(𝑠𝑛, 𝑡𝑛) > 0 for 𝑛 ∈ ℕ, then 𝑧(𝑠𝑛, 𝑡𝑛) > 0 for 𝑛 ∈ ℕ as well. We may assume 𝑡𝑛 ≠ 0 for 𝑛 ∈ ℕ and construct a function 𝜓 ∈ 𝐸 with 𝜓(𝑠𝑛, 𝑡𝑛) = 𝑛 ⋅ 𝑧(𝑠𝑛, 𝑡𝑛).

(6.15)

Since 𝜑 is a finite element in 𝐸, for some 𝑐𝜓 > 0 and all 𝐶 > 0 it holds that 𝜓(𝑠, 𝑡) ∧ 𝐶𝜑(𝑠, 𝑡) ≤ 𝑐𝜓 𝑧(𝑠, 𝑡) for all (𝑠, 𝑡) ∈ 𝐷.

(6.16)

For the points (𝑠𝑛, 𝑡𝑛) from (6.16), and with respect to (6.15), we obtain the inequality 𝑛𝑧(𝑠𝑛 , 𝑡𝑛) ≤ 𝑐𝜓 𝑧(𝑠𝑛, 𝑡𝑛), for infinitely many natural numbers; a contradiction. From the obtained characterization of finite elements in 𝐸 it is easily seen that 𝐸 satisfies the condition (ii), and that Φ1 (𝐸) is complete in 𝐸, i. e., M𝜑 = M holds. It is also clear that the set of all finite functions with supports in 𝐷 is a vector lattice of type (Σ). For the vector lattice 𝐸 the conditions (b), (i), and (ii) are satisfied. However, we will show that condition (a) does not hold, i. e., 𝐸 fails to be a vector lattice of type (Σ). Indeed, for any sequence (𝑧𝑛)𝑛∈ℕ of nonnegative functions in 𝐸, a function 𝑦 ∈ 𝐸 may be constructed for which lim

𝑘→∞

𝑧𝑛(0, 1 − 𝑘1 ) 𝑦(0, 1 − 1𝑘 )

= 0 for all 𝑛 ∈ ℕ.

The latter shows that 𝐸 cannot contain any sequence which satisfies the condition (Σ󸀠 ). The discrete functional 𝐹on 𝐸 vanishes on all finite functions in 𝐸. Hence MΦ ≠ M. Additionally, one observes that the spaces 𝐷 and M𝜑 are homeomorphic (see Theorem 8.20) and, moreover that M = M𝜑 ∪ {𝐹−1 (0)} holds. The following examples are based on a common construction. More exactly, we shall consider infinite matrices as real continuous functions on the set 𝑇 = ℕ × ℕ, which is equipped with discrete topology. For further investigation we need notations for several kinds of subsets of 𝑇. A subset of 𝑇 is called a row if it is of the kind {𝑠} × ℕ, where 𝑠 ∈ ℕ and is called a 𝜑-set, whether its intersection with each row is empty or finite. Later, at the end of Section 9.1, we will specify two more types of subsets of 𝑇. Denote for 𝜈 ∈ ℕ by 𝑒𝜈 the matrix

𝑒𝜈 = (𝑒(𝜈) 𝑖𝑗 )𝑖,𝑗∈ℕ

1 2𝜈 3𝜈 . . . 𝑛𝜈 . . . 1 2𝜈 3𝜈 . . . 𝑛𝜈 . . . ( .......................... ) = ( 1 2𝜈 3𝜈 . . . 𝑛𝜈 . . . ) , 1 1 1 ... 1 ... .. .. .. .. . ) . . ( .

(6.17)

where the first 𝜈 rows coincide with the first one and the remaining are rows with entries 1 at each position.

140 | 6 Topological characterization of finite elements Example 6.35. Example of an (𝑟)-complete vector lattice 𝐸 of type (Σ), in which Φ2 (𝐸) is not of type (Σ), (or, similarly, the subspace MΦ is not 𝜎-compact). Moreover, MΦ ≠ MΦ . At the same time this is a counterexample to Theorem 6.32, which shows that condition (i) (Φ2 (𝐸) is of type (Σ)) is not dispensable for MΦ to be closed in M. 𝐸 is now the collection of all matrices 𝑥 = (𝑥𝑖𝑗 )𝑖,𝑗∈ℕ for which numbers 𝜆 and 𝜈 exist such that 󵄨 󵄨 (𝜈) |𝑥| ≤ 𝜆𝑒𝜈 , i. e., 󵄨󵄨󵄨󵄨𝑥𝑖𝑗 󵄨󵄨󵄨󵄨 ≤ 𝜆𝑒𝑖𝑗 for 𝑖, 𝑗 ∈ ℕ. Under the natural (coordinatewise) algebraic operations and order 𝐸 is a vector lattice, where the conditions (a) and (b) are obviously satisfied. The finite elements in this vector lattice are characterized as follows. Lemma 6.36. An element 𝑥 = (𝑥𝑖𝑗 )𝑖,𝑗∈ℕ ∈ 𝐸 is finite if and only if its support, i. e., the set {(𝑖, 𝑗) ∈ 𝑇 : 𝑥𝑖𝑗 ≠ 0}, is a 𝜑-set. Proof. Indeed, since the sufficiency of the statement is clear, we will only prove the necessity. If 𝑥 ∈ Φ1 (𝐸), then for some 𝜈0 and each 𝜈 ∈ ℕ one finds a number 𝑐𝜈 > 0 such that 𝑒𝜈 ∧ 𝑛 |𝑥| ≤ 𝑐𝜈 𝑒𝜈0 for all 𝑛 ∈ ℕ. (6.18) For fixed 𝑖 ∈ ℕ, consider 𝜈 ∈ ℕ with 𝜈 > 𝜈0 and 𝜈 ≥ 𝑖. Then 𝜈 𝑒(𝜈) 𝑖𝑗 = 𝑗

(𝜈 )

and 𝑒𝑖𝑗 0 ≤ 𝑗𝜈0

𝑗 ∈ ℕ,

󵄨 󵄨 which together with (6.18) gives 𝑗𝜈 ∧ 𝑛 󵄨󵄨󵄨󵄨𝑥𝑖𝑗 󵄨󵄨󵄨󵄨 ≤ 𝑐𝜈 𝑗𝜈0 . Consequently, in the case 𝑥𝑖,𝑗 ≠ 0, one has the estimate 𝑗𝜈 ≤ 𝑐𝜈 𝑗𝜈0 , which is possible only if 𝑥𝑖𝑗 = 0 for 𝑗 ≥ 𝑗𝑖 , i. e., if the support of 𝑥 is a 𝜑-set. Observe that not every finite element in this vector lattice is a finite function. It is clear that the characterization of finite elements in 𝐸 immediately shows that each finite element is also totally finite and therefore possesses a finite element as one of its majorants. E. g., the characteristic function of the support of 𝑥 serves as such a majorant. So the condition (ii) is satisfied in 𝐸. We show that condition (i) does not hold, i. e., the subspace MΦ is not 𝜎-compact. For this purpose we establish that the ideal Φ2 (𝐸) (which coincides with the ideal Φ1 (𝐸)) is not of type (Σ). Let (𝑑𝜈 )𝜈∈ℕ with 𝑑𝜈 = (𝑑(𝜈) 𝑖𝑗 )𝑖,𝑗∈ℕ be an arbitrary sequence of positive elements in Φ2 (𝐸). Then for each 𝜈 ∈ ℕ, a number 𝑗𝜈 can be found such that 𝑑(𝜈) 𝑖𝑗 = 0 for 𝑗 > 𝑗𝜈 . Then the element 𝜔 = (𝜔𝜈𝑗 )𝜈,𝑗∈ℕ defined by 𝜔𝜈𝑗 = {

1, 0,

for for

𝑗 ≤ 𝑗𝜈 + 1 , 𝑗 > 𝑗𝜈 + 1

𝜈, 𝑗 ∈ ℕ

6.4 Examples |

141

is finite, however it is not majorized by any element of the kind 𝐶 𝑑𝜈 . Therefore, no sequence of Φ2 (𝐸) can satisfy the condition (Σ󸀠 ) in the vector lattice Φ2 (𝐸). The set Φ1 (𝐸) is easily seen to be complete in 𝐸, i. e., MΦ = M(𝐸). We show that MΦ ≠ M(𝐸) by obtaining a discrete functional on 𝐸 which vanishes on all finite elements. In this way we also provide the counterexamples referred to in Corollary 6.30 and Remark 6.31 in connection with the closedness of M𝜑 in M. Consider an ultrafilter U which contains all sets of the kind 𝑇 \ 𝑊, where the set 𝑊 is a union of a 𝜑-set with a finite number of rows. Since each function of 𝐸 is bounded on one of the sets of U for each 𝑥 ∈ 𝐸, the finite limit 𝑓(𝑥) = lim 𝑥 U

𝑥∈𝐸

exists. The functional 𝑓 is discrete and vanishes on the whole Φ1 (𝐸). Example 6.37. Example of a non (𝑟)-complete vector lattice 𝐹 of type (Σ) with Φ1 (𝐹) = Φ2 (𝐹), where Φ2 (𝐹) is vector lattice of type (Σ), but MΦ ≠ MΦ = M. At the same time it is a counterexample to Theorem 6.32, which shows that condition (b) (𝐸 is (𝑟)-complete) is not dispensable for MΦ to be closed in M. The vector lattice 𝐹 which is now needed is a sublattice of the vector lattice 𝐸 which was examined in the previous example and consists of all matrices whose elements are equal from some row on, i. e., 𝐹 = {𝑥 = (𝑥𝑖𝑗 )𝑖,𝑗∈ℕ ∈ 𝐸 : ∃ 𝑁𝑥 ∈ ℕ with 𝑥𝑖𝑗 = constant for 𝑖 ≥ 𝑁𝑥 , 𝑗 ∈ ℕ}. 𝐹 is a vector lattice of type (Σ) since the matrices 𝑒𝜈 ∈ 𝐹 for all 𝜈 ∈ ℕ, but 𝐹 is not 1 uniformly complete. The latter is understood if the matrix ∑∞ 𝑛=1 2𝑛 𝑥𝑛 is considered with 𝑥𝑛 = (𝑥(𝑛) 𝑖𝑗 )𝑖,𝑗∈ℕ ,

where 𝑥(𝑛) 𝑖𝑗 = {

1, 0,

if

𝑖=𝑗=𝑛 . otherwise

1 The partial sums of the series ∑∞ 𝑛=1 2𝑛 𝑥𝑛 compose a uniform Cauchy sequence (e. g., with the matrix 𝑦 = (𝑦𝑖𝑗 )𝑖,𝑗∈𝑁 as the regulator, where 𝑦𝑖𝑗 = 1 for all 𝑖, 𝑗 ∈ ℕ), which is not (𝑟)-convergent in 𝐹.

The finite elements in 𝐹 are described in the next lemma. Lemma 6.38. An element 𝑦 = (𝑦𝑖𝑗 )𝑖𝑗∈ℕ ∈ 𝐹 is finite if and only if 𝑦𝑖𝑗 = 0 holds, except a finite number of indices (𝑖, 𝑗) ∈ 𝑇. Proof. Indeed, the sufficient part is again clear. For the necessary part mention that 𝑒𝜈 ∈ 𝐹 for all 𝜈 ∈ ℕ. This shows that each finite element 𝑦 in 𝐹 is also finite in 𝐸, i. e., 𝑦 ∈ Φ1 (𝐸) ∩ 𝐹. It is also clear that any matrix belonging to Φ1 (𝐸) ∩ 𝐹 possesses only a finite number of nonzero entrances. From this characterization of the finite elements in 𝐹, the two facts immediately follow: Φ1 (𝐹) = Φ2 (𝐹); and Φ2 (𝐹) is a vector lattice of type (Σ). So the vector lattice 𝐹 satisfies

142 | 6 Topological characterization of finite elements the conditions (i) and (ii). The ideal Φ1 (𝐹) is complete in 𝐹, i. e., MΦ (𝐹) = M(𝐹). In order to show that MΦ (𝐹) ≠ M(𝐹) holds, we provide the functional 𝑓 defined by 𝑓(𝑦) = lim 𝑦𝑖1 , 𝑖→∞

where 𝑦 = (𝑦𝑖𝑗 )𝑖,𝑗∈ℕ ∈ 𝐹,

which vanishes on Φ1 (𝐹). Notice that this is also a counterexample to Corollary 6.30 with MΦ ≠ MΦ although Φ1 (𝐸) = Φ2 (𝐸). Example 6.39. Example of an (𝑟)-complete vector lattice 𝐸 of type (Σ), where Φ2 (𝐸) is vector lattice of type (Σ), but Φ1 (𝐸) ≠ Φ2 (𝐸) and MΦ ≠ MΦ . At the same time it is a counterexample to Theorem 6.32, which shows that condition (ii) (Φ1 (𝐸) = Φ2 (𝐸)) is not dispensable for MΦ to be closed in M. We use the already introduced Kaplanski vector lattice K (see Example 3.4, p. 23) for the required example. In Sections 3.1, 6.2, and in Example 3.4 we already described this vector lattice in detail. Here we repeat the facts that were already pointed out: K is an (r)-complete vector lattice of type (Σ), where the sequence 𝜈

𝑒𝜈 (𝑡) = ∑ 𝑘=1

1 , |𝑘𝑡 − 1|

1 1 1 𝑡 ∈ 𝑇 = [−2, 2] \ {1, , , , . . .}, 2 3 4

𝜈 = 1, 2, . . .

satisfies the condition (Σ󸀠 ). From the characterization (a), (b), and (c) of finite and totally finite elements in K (see p. 23), it is clear that Φ2 (K) is a vector lattice of type (Σ), i. e., MΦ = MΦ (K) is 𝜎-compact, but Φ1 (K) ≠ Φ2 (K). One has MΦ = M(K) since the ideal Φ1 (K) is complete in K. To show that MΦ ≠ M(K), it suffices to indicate a discrete functional which vanishes on Φ1 (K). Such a one is the evaluation at point 𝑡 = 0. The last example enables us to provide the example which was referred to after Corollary 6.18. Example 6.40. A vector lattice 𝐸0 in which a maximal ideal from Φ1 (𝐸0 ) cannot be extended to a maximal ideal in 𝐸0 . We keep all notations used in the previous Example 6.39. For 𝐸0 we take the linear sublattice of the vector lattice 𝐸 of that example, which consists of all functions 𝑥 ∈ 𝐸 that satisfy the condition |𝑥(𝑡) − 𝑥(0)| < +∞. sup |𝑡| −2 0, the set 𝐺𝑥,𝜀 = {𝑡 ∈ 𝑄 : 𝑥(𝑡) > 𝜀} is open in 𝑄 and one has 𝐾𝑥,2𝜀 ⊂ 𝐺𝑥,𝜀 ⊂ 𝐾𝑥,𝜀 . (7.1) The condition (⋆) implies 𝑄 =

⋃

(7.2)

𝐾𝑥,𝜀 .

00

The second statement follows immediately from (7.1) and (7.2), after the compactness of 𝐾𝑥,𝜀 is established. We assume by way of contradiction that for some 𝑥0 ∈ 𝐸(𝑄) and 𝜀0 > 0 the set 𝐾0 = 𝐾𝑥0 ,𝜀0 is not compact. The collection 𝐾0 \ 𝐾 for any compact subset 𝐾 of 𝑄 is the basis of certain filter U in 𝑄 which is finer than the filter containing the complements of all compact subsets. We have therefore U → ∞, i. e., for each compact subset 𝐾 ⊂ 𝑄 there is a set 𝑈 ∈ U with 𝑈 ⊂ 𝑄 \ 𝐾. Since, by assumption 𝑥0 (𝑡) 󳨀→ 0, the 𝑡→∞

limit lim 𝑥0 exists and is equal to zero. However, 𝑥0 (𝑡) ≥ 𝜀0 > 0 on any set 𝐾0 \ 𝐾. U

Corollary 7.4. If (𝐸(𝑄), 𝑖) is (⋆)𝑐0 -representation of a vector lattice 𝐸, then the space 𝑄 is locally compact. Let 𝐸(𝑄) be a (⋆)-representation of the vector lattice 𝐸, i. e., there exist some Hausdorff space 𝑄, a vector lattice 𝐸(𝑄) ⊆ 𝐶(𝑄) and a Riesz isomorphism 𝑖 : 𝐸 → 𝐸(𝑄). Due to the condition (⋆), for any point 𝑡 ∈ 𝑄 the sets 𝑀𝑡 = {𝑥 ∈ 𝐸(𝑄) : 𝑥(𝑡) = 0} and

𝑖−1 (𝑀𝑡 )

are maximal ideals in the vector lattices 𝐸(𝑄) and 𝐸 respectively.

146 | 7 Representations of vector lattices and their properties Definition 7.5. The map 𝜅 : 𝑄 → M(𝐸) defined by 𝜅(𝑡) = 𝑖−1 (𝑀𝑡 ) is called the standard map. The continuity of the standard map and some other properties of 𝜅 (depending on 𝑖) will be proved next. Under additional conditions, the spaces 𝑄 and M(𝐸) might even be canonically homeomorphic; see Theorem 7.12. The standard map will be used for further classification of representations for vector lattices by means of continuous functions. Observe first that 𝑥(𝑡) = 0 ⇐⇒ 𝑥 ∈ 𝑀𝑡 ⇐⇒ 𝑖−1 𝑥 ∈ 𝜅(𝑡).

(7.3)

Theorem 7.6. The standard map 𝜅 : 𝑄 󳨀→(M, 𝜏ℎ𝑘 ) is continuous. Proof. Let 𝐴 be an arbitrary subset of 𝑄 and 𝑡0 a closure point of 𝐴. If 𝜅(𝑡0 ) is not a closure point of the set 𝜅(𝐴), then 𝜅(𝑡0 ) ⊉ ⋂𝑀∈𝜅(𝐴) 𝑀. Consequently, an element 𝑥0 ∈ 𝐸 exists with 𝑥0 ∈ 𝑀 for all 𝑀 ∈ 𝜅(𝐴), and 𝑥0 ∉ 𝜅(𝑡0 ), i. e., 𝑥0 (𝑡0 ) ≠ 0. There is neighborhood 𝑈0 of the point 𝑡0 in 𝑄, where the function 𝑥0 does not vanish. If a point 𝑡 now belongs to the set 𝐴, then 𝑡 = 𝜅−1 (𝑀) for some 𝑀 ∈ 𝜅(𝐴). Since 𝑥0 ∈ 𝑀 for all 𝑀 ∈ 𝜅(𝐴), we have 𝑥0 (𝑡) = 0 for all 𝑡 ∈ 𝐴. However, 𝑈0 ∩ 𝐴 ≠ 0. Hence 𝑥0 (𝑡󸀠 ) ≠ 0 and 𝑥0 (𝑡󸀠 ) = 0 for all points 𝑡󸀠 ∈ 𝑈0 ∩ 𝐴. Proposition 7.7. Let (𝐸(𝑄), 𝑖) be a (⋆)-representation of the vector lattice 𝐸. The following properties are equivalent: (1) the standard map 𝜅 is injective; (2) for an arbitrary (not necessarily ordered) pair of points 𝑡, 𝑡󸀠 ∈ 𝑄, 𝑡 ≠ 𝑡󸀠 , a function 𝑥 ∈ 𝐸(𝑄) exists, which vanishes at one point and is different from zero at the other one; (3) 𝐸(𝑄) strongly separates the points of 𝑄. Proof. (1) 󳨐⇒ (2) Due to the injectivity of 𝜅 for the points 𝑡, 𝑡󸀠 ∈ 𝑄 with 𝑡 ≠ 𝑡󸀠 , the sets 𝑀𝑡 and 𝑀𝑡󸀠 are different, and so the ideal 𝑀𝑡 contains a function 𝑥 ∈ 𝐸(𝑄) which does not belong to 𝑀𝑡󸀠 . It follows 𝑥(𝑡) = 0 and 𝑥(𝑡󸀠 ) ≠ 0. (2) 󳨐⇒ (3) In view of the maximality of the ideals 𝑀𝑡 and 𝑀𝑡󸀠 behind the function 𝑥 obtained in the first part of the proof, there is another function 𝑦 ∈ 𝑀𝑡󸀠 which does not belong to 𝑀𝑡 . Without loss of generality we may assume 𝑥(𝑡󸀠 ) = 𝑦(𝑡) = 1. This shows that the ordered pair (𝑡, 𝑡󸀠 ) of points is strongly separated by means of functions of 𝐸(𝑄). (3) 󳨐⇒ (1) For the points 𝑡 ≠ 𝑡󸀠 there are functions 𝑥,̃ 𝑦̃ ∈ 𝐸(𝑄) such that ̃ = 𝑦(𝑡 ̃ 󸀠) = 0 and 𝑥(𝑡) i. e., 𝑀𝑡 ≠ 𝑀𝑡󸀠 .

̃ 󸀠 ) = 𝑦(𝑡) ̃ = 1, 𝑥(𝑡

7.1 A classification of representations and the standard map | 147

Definition 7.8. A representation (𝐸(𝑄), 𝑖) of a vector lattice 𝐸 is called – an e-representation if the standard map 𝜅 : 𝑄 󳨀→ M(𝐸) is surjective; – an E-representation if the standard map 𝜅 is bijective. Obviously, any E-representation is an e-representation, and the latter is a (⋆)representation. However, not all e-representations are E-representations. It is convenient to provide the corresponding Counterexample 9.25 in Section 9.4. The notion of an E-representation provides the opportunity for indicating two properties which are necessary and sufficient for a (⋆)-representation to actually be a representation on the space (M(𝐸), 𝜏ℎ𝑘 ). This important result follows the two propositions proved first. Proposition 7.9. If (𝐸(𝑄), 𝑖) is a completely regular E-representation of a vector lattice 𝐸, then the standard map 𝜅 is a homeomorphism. Proof. According to Theorem 7.6, the standard map 𝜅 is continuous, hence it suffices to show that 𝜅 is an open mapping. For that, let 𝐺 be an open subset of 𝑄 and 𝑡0 ∈ 𝐺. We assume that 𝜅(𝑡0 ) is not an interior point of 𝜅(𝐺). Then 𝜅(𝑡0 ) is a closure point of the set M \ 𝜅(𝐺), i. e., 𝜅(𝑡0 ) ⊃ ⋂ 𝑀. (7.4) 𝜅−1 (𝑀)∉𝐺

The relation 𝑥 ∈ 𝑀 for 𝜅−1 (𝑀) ∉ 𝐺 is equivalent to 𝑥(𝑡) = 0 for all 𝑡 ∈ 𝐺. Then (7.4) implies that such 𝑥 must belong to the maximal ideal 𝜅(𝑡0 ), which means 𝑥(𝑡0 ) = 0. Therefore we have shown that whenever a function of 𝐸(𝑄) vanishes outside of 𝐺 it also vanishes at the point 𝑡0 . This is a contradiction of the complete regularity of the representation (𝐸(𝑄), 𝑖). Corollary 7.10. Under the conditions of the proposition, the space (M, 𝜏ℎ𝑘 ) is completely regular. The proposition has a converse. Proposition 7.11. Let (𝐸(𝑄), 𝑖) be a (⋆)-representation of a vector lattice 𝐸. If the standard map 𝜅 is a homeomorphism, then (𝐸(𝑄), 𝑖) is a completely regular E-representation. Proof. Let 𝐹 ⊂ 𝑄 be a closed set and 𝑡0 a point with 𝑡0 ∉ 𝐹. By assumption 𝜅(𝑡0 ) ∉ 𝜅(𝐹) = 𝜅(𝐹), i. e., 𝜅(𝑡0 ) ⊅ ⋂ 𝜅(𝑡). 𝑡∈𝐹

Hence there is an element 𝑧 ∈ ⋂𝑡∈𝐹 𝜅(𝑡) which does not belong to 𝜅(𝑡0 ). With respect to (7.3), this means 𝑥0 (𝑡) = 0 for all 𝑡 ∈ 𝐹

and 𝑥0 (𝑡0 ) ≠ 0,

where 𝑥0 = 𝑖−1 (𝑧). This shows the complete regularity of the representation, which is obviously an E-representation of 𝐸.

148 | 7 Representations of vector lattices and their properties By combining the results of the two last propositions we get a result on the representation of a vector lattice on its space of maximal ideals. Theorem 7.12. Let (𝐸(𝑄), 𝑖) be a (⋆)-representation of a vector lattice 𝐸. The standard map 𝜅 is a homeomorphism of 𝑄 onto (M(𝐸), 𝜏ℎ𝑘 ) if and only if the following two conditions hold: (1) (𝐸(𝑄), 𝑖) is an E-representation; (2) (𝐸(𝑄), 𝑖) is a completely regular representation. Remark 7.13. If a vector lattice 𝐸 possesses two different completely regular Erepresentations (𝐸(𝑄), 𝑖) and (𝐸(𝑆), 𝑗), then the topological spaces 𝑄 and 𝑆 are homeomorphic.

7.2 Vector lattices of type (Σ) and their representations Next we define further classes of qualified representations of vector lattices and consider representations for vector lattices of type (Σ). For this class we are able to formulate conditions under which different representations exist . Definition 7.14. The representation (𝐸(𝑄), 𝑖) of the vector lattice 𝐸 is called 𝜎representation if the topological space 𝑄 is 𝜎-compact, i. e., is a countable union of compact subsets of 𝑄. A representation is called b-representation if it consists of bounded continuous functions. Each 𝑐0 -representation is also a b-representation. Vector lattices of type (Σ) introduced in Section 2.3 will be the main object of our investigation in the present chapter and in Chapter 9. The vector lattices of majorizing functions and the vector lattices of slowly growing functions, introduced and studied in Section 8.2, are vector lattices of type (Σ) by definition. The vector lattice in Example 6.34 fails to be of type (Σ). The properties obtained for the class of vector lattices of type (Σ) in this section essentially complete the properties which already have been established in Section 6.3. This paves the way for the representation theory for vector lattices of type (Σ), developed almost exhaustively in this section. Notice that the necessary condition for representability mentioned at the beginning of Section 7.1 is satisfied, as established in Theorem 5.8 (see also Proposition 5.13), i. e., in any vector lattice 𝐸 of type (Σ) the collection Δ(𝐸) of all discrete functionals is total. We now add another corollary to Theorem 5.8. Corollary 7.15. If in a vector lattice of type (Σ) a set is sufficient then (due to Corollary 5.12), it is also complete.

7.2 Vector lattices of type (Σ) and their representations

| 149

The next theorem is a supplement to Theorem 7.3. For a vector lattice of type (Σ) consisting of continuous functions on 𝑄 some properties of the space 𝑄 will be revealed which are important further on. Theorem 7.16. Let 𝐸(𝑄) be a vector lattice of type (Σ) which consists of continuous functions on the topological space 𝑄. Let (𝑒𝜈)𝜈∈ℕ be a sequence which satisfies the condition (Σ󸀠) in 𝐸(𝑄). Then the following statements hold: (1) if 𝑄 is compact then a point 𝑡0 ∈ 𝑄 exists at which all functions of 𝐸(𝑄) vanish; (2) if 𝐸(𝑄) satisfies the conditions (⋆) and (𝑐0 ), then the topological space 𝑄 is locally compact and 𝜎-compact. Proof. (1) From the contrary. Suppose for each point 𝑡 ∈ 𝑄, a number 𝜈𝑡 with 𝑒𝜈𝑡 (𝑡) > 0 can be found. The open sets 𝑈𝑡 = {𝑠 ∈ 𝑄 : 𝑒𝜈𝑡 (𝑠) > 0} form an open covering for 𝑄. 𝑁 Then for some 𝑡1 , 𝑡2 , . . . , 𝑡𝑁 one has 𝑄 ⊂ ⋃𝑖=1 𝑈𝑡𝑖 . For sufficiently large 𝑙 there holds 𝑒𝑙 (𝑡) > 0 for all 𝑡 ∈ 𝑄, and hence the functions 𝑒𝜈 for 𝜈 > 𝑙 majorize each other. That means 𝐸(𝑄) is a vector lattice of bounded elements, which is excluded (by definition) if 𝐸(𝑄) is a vector lattice of type (Σ). (2) The presumptions lead to 𝑄 = ⋃∞ 𝜈,𝑛=1 𝐾𝜈, 1 , where the compactness of the sets 𝑛

𝐾𝜈, 1 = {𝑡 ∈ 𝑄 : 𝑒𝜈 (𝑡) ≥ 𝑛1 } has already been established in Theorem 7.3 (1). 𝑛

Remark 7.17. Theorem 7.3 and the second part of the previous theorem show that the topological space 𝑄 is locally compact whenever in 𝐸(𝑄) there is a set 𝐴 with the following properties: (a) for any point 𝑡 ∈ 𝑄 there is a function 𝑥 ∈ 𝐴 with 𝑥(𝑡) ≠ 0; (b) for each function 𝑥 ∈ 𝐴 one has 𝑥(𝑡) 󳨀→ 0. 𝑡→∞

Moreover, if the set 𝐴 is countable, then 𝑄 is also 𝜎-compact. Corollary 7.18. A vector lattice of type (Σ) cannot possess any (⋆)-representation on a compact space. Corollary 7.19. If (𝐸(𝑄), 𝑖) is a (⋆)𝑐0 -representation of a vector lattice 𝐸 of type (Σ), then the space 𝑄 is locally compact (cf. Corollary 7.4 to Theorem 7.3) and 𝜎-compact. Hence (𝐸(𝑄), 𝑖) is a 𝜎-representation on the locally compact space 𝑄. In connection with Theorem 7.3, the last theorem convinces that it is quite natural to consider representations on spaces more general than compact ones, e. g., on locally compact topological spaces, where in case of vector lattices of type (Σ) the 𝜎compactness may be required additionally. Now we address the question of the existence of representations for vector lattices of type (Σ). The considerations will be continued and completed in Section 9.2 First we present a general construction (scheme) of isomorphic classes of vector lattices of type (Σ) consisting of continuous functions. This procedure will be applied several times later on.

150 | 7 Representations of vector lattices and their properties Let 𝐸(𝑄) be a vector lattice of type (Σ) of continuous functions on a 𝜎-compact topological space 𝑄 in which the sequence (𝑒𝜈 )𝜈∈ℕ of functions satisfies the condition (Σ󸀠 ). Let 𝑄 = ⋃∞ 𝑛=1 𝐾𝑛 , where 𝐾𝑛 are compact subsets of 𝑄 for all 𝑛 ∈ ℕ. (1) Put 1 𝛼𝜈 = 1 + max 𝑒𝜈(𝑡) and 𝜆 𝜈 = 𝜈 . 𝑡∈𝐾𝜈 2 𝛼𝜈 The function 𝑎(𝑡) = 1+∑∞ 𝜈=1 𝜆 𝜈 𝑒𝜈 (𝑡) is continuous on each set 𝐾𝑛 for 𝑛 ∈ ℕ. However, it may not be continuous on the whole space 𝑄. We strengthen the topology on 𝑄 in order to force the function 𝑎 to be continuous on 𝑄. This will be done as follows: a subset 𝐺 ⊂ 𝑄 is declared to be open (in the new topology 𝜏1 of 𝑄) if for any 𝑛 the set 𝐾𝑛 ∩𝐺 is open in 𝐾𝑛 . The topology 𝜏1 defined in that way on 𝑄 is stronger than the original one, and the function 𝑎 is continuous on 𝑄1 , where 𝑄1 = (𝑄, 𝜏1 ) is the set 𝑄 equipped with the topology 𝜏1 . Observe that the infimum of the function 𝑎 on 𝑄 is strongly positive, hence each function 𝑥 ∈ 𝐸(𝑄) may be divided by 𝑎. Thus the obtained set of continuous functions 𝑥 𝐸(𝑄1 ) = { : 𝑥 ∈ 𝐸(𝑄) } 𝑎 is an isomorphic vector lattice to 𝐸(𝑄) of type (Σ), which consists of bounded functions. (2) Let now 𝑄 = (𝑄, 𝜏) be a topological space and 𝐸(𝑄) a vector lattice of type (Σ) consisting of bounded continuous functions on 𝑄. In general, the space 𝑄 may not be completely regular – the reason for replacing the topology 𝜏 by a stronger, completely regular one 𝜏. Denote the pair (𝑄, 𝜏) by 𝑄 and the set of functions 𝐸(𝑄) considered on the space 𝑄 by 𝐸(𝑄). The latter is again a vector lattice of bounded continuous functions. Consequently, by using the isomorphism between the vector space 𝐶𝑏(𝑄) of all bounded continuous functions, and the space 𝐶(𝛽𝑄) of all continuous functions on the Stone–Čech compactification 𝛽𝑄 of 𝑄, the set 𝐸(𝑄) is isomorphic to some subset 𝐸0 (𝛽𝑄) ⊂ 𝐶(𝛽𝑄); see [49]. Thus 𝐸0 (𝛽𝑄) is a vector lattice of type (Σ) of continuous functions on the compact space 𝛽𝑄. (3) According to Theorem 7.16 (1), the set 𝐹 = { 𝑡 ∈ 𝛽𝑄 : 𝑥(𝑡) = 0 for all 𝑥 ∈ 𝐸0 (𝛽𝑄) } is not empty. Denote now the restrictions of all functions of 𝐸0 (𝛽𝑄) onto the set 𝑄2 = 𝛽𝑄\𝐹 by 𝐸(𝑄2 ); the latter set then satisfies the condition (⋆). From the observation that for each function 𝑥 ∈ 𝐸(𝑄2 ) and any 𝜀 > 0 a neighborhood 𝑈 of 𝐹 in 𝛽𝑄 exists such that 𝑥(𝑡) ∈ (−𝜀, 𝜀) for 𝑡 ∈ 𝑈, we conclude that 𝐸(𝑄2 ) also satisfies the condition (𝑐0 ). According to Theorem 7.16 (2), the space 𝑄2 is then locally compact and 𝜎-compact. The scheme provided is easily seen to be a generalization of the approach used in Section 6.4 for the representation of some particular vector lattices of infinite matrices. It also explains, on the one hand, the existence of isomorphic mappings of a vector lattice of type (Σ) of continuous functions on a 𝜎-compact space 𝑄 onto a vector lattice of bounded continuous functions, and on the other hand, the existence of isomorphic mappings from vector lattices of the last class onto a vector lattice 𝐸(𝑄2 ) of type (Σ) of

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continuous functions on a locally compact 𝜎-compact space 𝑄2 , where in 𝐸(𝑄2 ) even the conditions (⋆) and (𝑐0 ) are satisfied: 𝑖1

𝑖2

𝐸(𝑄) 󳨀→ 𝐸(𝑄1 ) 󳨀→ 𝐸(𝑄2 ). Remark 7.20. The conditions posed on the last vector lattice may still be strengthened. Additionally, one can demand that the vector lattice 𝐸(𝑄2 ) separates the points of 𝑄2 . In order to see this, it is sufficient to factorize the space 𝛽𝑄 with respect to the equivalence relation 𝜉, where for 𝑡, 𝑡󸀠 ∈ 𝛽𝑄 the relation 𝑡 𝜉 𝑡󸀠 holds if 𝑥(𝑡) = 𝑥(𝑡󸀠 ) for all functions 𝑥 ∈ 𝐸0 (𝛽𝑄). Theorem 7.21. Let 𝐸 be a vector lattice of type (Σ). The following statements are equivalent: (1) a Riesz norm exists on 𝐸; (2) a 𝜎-representation exists; (3) a b-representation exists; (4) a (⋆)𝑐0 -representation exists on a locally compact 𝜎-compact space; (5) the vector lattice 𝐸 is contained as a linear sublattice in some vector lattice of bounded elements. Proof. (1) ⇒ (2). Let ‖⋅‖0 be a monotone norm on 𝐸 and denote by 𝑍 the Banach completion of the normed space (𝐸, ‖⋅‖0 ). It is clear that 𝑍 is a Banach lattice ([9, Theorem 12.2]) and in 𝑍 the element ∞

1 = ∑ 𝜈=1

2𝜈

𝑒𝜈 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑒𝜈󵄩󵄩0

exists, where (𝑒𝜈 )𝜈∈ℕ is a sequence of elements which satisfies the condition (Σ󸀠 ) in 𝐸. Consider in 𝑍 the vector lattice of bounded elements 𝑍0 = {𝑧 ∈ 𝑍 : |𝑧| ≤ 𝜆1} equipped with its “natural” 𝑢-norm ‖𝑧‖ = inf{𝜆 : |𝑧| ≤ 𝜆1}. There exists a total set of discrete functionals on 𝑍0 which are continuous with respect to the norm ‖⋅‖. Denote by Δ the set of all discrete functionals on 𝐸 which satisfy the condition 𝑘

𝑓(1) = lim ∑ 𝑘→∞

𝜈=1

1 󵄩 󵄩 𝑓(𝑒𝜈 ) ≤ 1. 2𝜈 󵄩󵄩󵄩𝑒𝜈 󵄩󵄩󵄩0

󵄨 󵄨 Since 󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨 ≤ 𝑓(|𝑥|) ≤ 𝑓(‖𝑥‖ 1) = ‖𝑥‖ 𝑓(1) for any 𝑥 ∈ 𝐸, and 𝑓 ∈ Δ, all functionals of Δ are continuous with respect to the norm ‖⋅‖. The set Δ contains the restrictions on 𝐸 of the total set of discrete functionals on 𝑍0 , which are continuous with respect to the norm ‖⋅‖ and therefore Δ is a total set on 𝐸 on its own. Consider now the elements of 𝐸 as functions given on Δ, where the value of a function 𝑥 ∈ 𝐸 at the point 𝑓 ∈ Δ is

152 | 7 Representations of vector lattices and their properties 𝑓(𝑥). If Δ is equipped with weak topology 𝜎((𝐸, ‖⋅‖)󸀠 , 𝐸), then each such function 𝑥(𝑓) is continuous. Denote this collection by 𝐸(Δ) and define the algebraic and order operations pointwise. The map 𝑖 : 𝐸 󳨀→ 𝐸(Δ) is a Riesz isomorphism. The set Δ ⊂ (𝐸, ‖⋅‖)󸀠 is weakly closed and the polar sets 󵄨 󵄨 𝑆∘𝑛 = {𝑓 ∈ (𝐸, ‖⋅‖)󸀠 : 󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨 ≤ 1, 𝑥 ∈ 𝑆𝑛} of the sets 𝑆𝑛 = {𝑥 ∈ 𝐸 : ‖𝑥‖ ≤ 𝑛1 } are weakly compact. Hence all sets 𝑆∘𝑛 ∩ Δ are ∞ ∘ 󸀠 ∘ weakly compact. Since ⋃∞ 𝑛=1 𝑆𝑛 = 𝐸 implies Δ = ⋃𝑛=1 (𝑆𝑛 ∩ Δ), we conclude that Δ is a 𝜎-compact space and (𝐸(Δ), 𝑖) is a 𝜎-representation of 𝐸. The proof of the implications (2) ⇒ (3) and (3) ⇒ (4) follows the scheme (1)–(3) which was specified before the theorem. (4) ⇒ (5). We identify the vector lattice 𝐸 with its assumed (⋆)𝑐0 -representation 𝐸(𝑄). As the vector lattice of bounded elements into which 𝐸 is embedded serves the vector lattice of all bounded continuous functions on 𝑄. (5) ⇒ (1). Let 𝐸 be contained in the vector lattice of bounded elements 𝑌 with the fixed order unit 𝑢. Then the 𝑢-norm ‖𝑥‖ = inf {𝜆 : |𝑥| ≤ 𝜆𝑢} is monotone on 𝐸. Not in any vector lattice of type (Σ) does a Riesz norm exist, as the next example shows. Example 7.22. An example of a Dedekind complete vector lattice of type (Σ) which does not possess any Riesz norm. Consider the set 𝑇 = [0, 1] as the index set of all positive increasing sequences of numbers. To each point 𝑡 ∈ 𝑇 there is assigned the sequence (𝑡) (𝑡) 𝜉𝑡 = (𝑛(𝑡) 1 , 𝑛2 , . . . , 𝑛𝑘 , . . . )

(𝑡) (𝑡) with 0 ≤ 𝑛(𝑡) 1 ≤ 𝑛2 ≤ ⋅ ⋅ ⋅ 𝑛𝑘 ⋅ ⋅ ⋅ .

Define on 𝑇 the functions 𝑒𝜈 by 𝑒𝜈 (𝑡) = 𝑛(𝑡) 𝜈 ,

for 𝑡 ∈ 𝑇 and 𝜈 = 1, 2, . . . .

The vector lattice 𝐸 = 𝐸(𝑇) is now defined to consist of all functions 𝑥(𝑡) such that its modulus is majorized by one of the functions 𝑒𝜈 , i. e., |𝑥(𝑡)| ≤ 𝜆𝑒𝜈 (𝑡), 𝑡 ∈ 𝑇 for some 𝜆 depending on 𝑥. Then it is clear that 𝐸 is Dedekind complete vector lattice of type (Σ). For each sequence (𝜇𝜈 )𝜈∈ℕ of positive numbers in 𝐸 there holds the relation inf 𝑛 𝜇𝑛 𝑒𝑛 > 0. This follows from the fact that there is point 𝑡0 in 𝑇 for which 𝑛𝜈(𝑡0 ) 𝜇𝜈 ≥ 1, 𝜈 ∈ ℕ. The latter implies {1, 𝑡 = 𝑡0 𝜇𝜈 𝑒𝜈 ≥ { . inf 𝜈 0, 𝑡 ≠ 𝑡0 { In order to conclude that there is no monotone norm in 𝐸, it suffices to show that 𝐸 is not embeddable into a vector lattice of bounded elements; cf. Theorem 7.21. Indeed, if 𝐸

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were isomorphically contained as a vector sublattice in some vector lattice of bounded elements 𝑌 with the order unit 𝑦0 , then there would be numbers 𝜆 𝜈 for 𝜈 ∈ ℕ with 𝑒𝜈 ≤ 𝜆 𝜈 𝑦0 . The sequence ( 21𝜈 𝑒𝜈)𝜈∈ℕ (𝑜)-converges in 𝑌 to zero. Since inf 𝜈 2𝜈1𝜆 𝑒𝜈 > 0 in 𝐸, and 𝜈 the infimum of this set in 𝑌 cannot be smaller than in 𝐸, we arrive at a contradiction to the (𝑜)-convergence to zero of the sequence in 𝑌. Remark 7.23. If ‖⋅‖ is a Riesz norm on a vector lattice, then it is possible that no maximal ideal is closed with respect to this norm. The corresponding Example 9.26 will be provided in Section 9.4. Remark 7.24. Let 𝐸(𝑄) be a vector lattice of continuous functions which satisfies the condition (𝑐0 ). If the finite functions belonging to 𝐸(𝑄) strongly separate the points of 𝑄, i. e., the condition (𝛼) (see Definition 8.1 below) holds, then a maximal ideal is closed with respect to the supremum norm if and only if it is of the kind {𝑥 ∈ 𝐸(𝑄) : 𝑥(𝑡0 ) = 0} for some point 𝑡0 ∈ 𝑄. Proof. The sufficiency is clear. For the remaining part, of necessity, we mention that the condition guarantees that the collection of the finite functions from 𝐸(𝑄) is dense in 𝐸(𝑄) with respect to the supremum norm. Then it is clear that a closed maximal ideal cannot contain all finite functions of 𝐸(𝑄). The proof will be completed by means of Theorem 8.9, which will be provided in Section 8.1, see Remark 8.10 (2). If a vector lattice possesses a 𝑐0 e-representation (𝐸(𝑄), 𝑖), then it is natural to consider the supremum norm on 𝐸(𝑄). With respect to this (obviously Riesz) norm, any maximal ideal is closed. For vector lattices of the type (Σ) some inverse result is true. Theorem 7.25. Let 𝐸 be a vector lattice of type (Σ) in which the sequence (𝑒𝜈)𝜈∈ℕ satisfies the condition (Σ󸀠 ). The following statements are equivalent: (1) a Riesz norm exists on 𝐸 such that all maximal ideals are closed with respect to it; (2) a norm exists on 𝐸 such that all maximal ideals are closed with respect to it; (3) a 𝜎e-representation exists; (4) a be-representation exists; (5) a 𝑐0 e-representation exists; (6) a sequence of numbers 𝜆 𝜈 > 0, 𝜈 ∈ ℕ exists, such that the value 𝑐𝑓 = sup 𝜈∈ℕ

𝑓(e𝜈 ) 𝜆𝜈

is finite for each discrete functional 𝑓. Proof. (1) ⇒ (2) is trivial. (2) ⇒ (3). Let ‖⋅‖ be a norm on 𝐸 with the corresponding properties. Then all discrete functionals are continuous. Introduce for 𝜈, 𝑛 ∈ ℕ the following sets 𝐹𝜈 = {𝑓 ∈ Δ(𝐸) : 𝑓(𝑒𝜈 ) = 1}; 𝑆𝑛 = {𝑥 ∈ 𝐸 : ‖𝑥‖ ≤ 1𝑛 }; 󵄨 󵄨 𝑆∘𝑛 = {𝑓 ∈ 𝐸󸀠 : 󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨 ≤ 1, 𝑥 ∈ 𝑆𝑛}.

154 | 7 Representations of vector lattices and their properties With respect to the weak∗ -topology 𝜎(𝐸󸀠 , 𝐸), the sets 𝐹𝜈 , 𝜈 ∈ ℕ are closed and 𝑆∘𝑛, 𝑛 ∈ ℕ compact sets. Consequently, with respect to this topology the set 𝑆 = ⋃𝜈,𝑛∈ℕ (𝐹𝜈 ∩ 𝑆∘𝑛 ) is a 𝜎-compact space. By means of 𝑥(𝑓) = 𝑓(𝑥), (𝑓 ∈ 𝑆) we consider the elements of 𝐸 as functions on 𝑆. The collection of all those functions together with the map 𝑥 󳨃→ 𝑥(𝑓) delivers a 𝜎-representation of 𝐸. For the discrete functional 𝑓𝑀 corresponding to 𝑀 ∈ M(𝐸), a number 𝜈0 exists such that 𝑓(𝑒𝜈0 ) ≠ 0. Then it is clear that the functional 𝑓󸀠 = 𝑓 (𝑒1 ) 𝑓𝑀 belongs to 𝑆. It follows that the obtained representation is 𝑀

𝜈0

a (⋆)-representation and the standard map 𝜅 : 𝑆 → M(𝐸) is surjective. Consequently, the representation is a 𝜎e-representation. (3) ⇒ (4) follows from the general scheme for constructing an isomorphic vector lattice of bounded continuous functions starting with a vector lattice of continuous functions on some 𝜎-compact space (step (1)). It is easy to see that the surjectivity of the standard map 𝜅 is preserved. (4) ⇒ (5). Let (𝐸(𝑆), 𝑖0 ) a be-representation of 𝐸. Without restriction of generality, the topological space 𝑆 can be assumed to be completely regular. Then 𝐸(𝑆) is isomorph to some subset 𝐸0 (𝛽𝑆) ⊂ 𝐶(𝛽𝑆), where 𝛽𝑆 denotes the Stone–Čech compactification of 𝑆. In 𝛽𝑆, by means of the equivalence relation 𝜉, we move on to the quotient space 𝛽𝑆/𝜉 and denote the latter by 𝑆󸀠 . Now to the functions of 𝐸0 (𝛽𝑆 assign the functions which canonically occur on 𝑆󸀠 , see [31, §§ 9, 10]. Denote them by 𝐸(𝑆󸀠 ). Let 𝑞∞ the quotient class of the set {𝑠 ∈ 𝛽𝑆 : 𝑥(𝑠) = 0 for all 𝑥 ∈ 𝐸0 (𝛽𝑆)}. Denote by 𝑇 the set 𝑆󸀠 \ {𝑞∞ } and by 𝐸(𝑇) all restrictions of the functions from 𝐸(𝑆󸀠 ) onto 𝑇. This way, −1 an isomorphic mapping 𝑖1 : 𝐸(𝑆) → 𝐸(𝑇) is constructed. We now put 𝑖 = 𝑖−1 0 ∘ 𝑖1 . The map 𝛾 : 𝑇 → M(𝐸), defined by 𝛾(𝑡) = 𝑖(𝛿𝑡−1 (0)), is the standard map (from 𝑇 to M). Since the original representation in this part of the proof was an e-representation, for any 𝑀 ∈ M there is point 𝑠 ∈ 𝑆, and further a quotient class 𝑡𝑠 ∈ 𝑇 such that 𝛾(𝑡𝑠 ) = 𝑀. Hence the pair (𝐸(𝑇), 𝑖1 ∘ 𝑖0 ) is an e-representation. At the same time it is a 𝑐0 -representation, which follows from the general scheme. (5) ⇒ (6). If (𝐸(𝑆), 𝑖) is a 𝑐0 e-representation, then the surjectivity of the standard map 𝜅 : 𝑆 → M(𝐸) implies that for any discrete functional 𝑓 there are a point 𝑠𝑓 ∈ 𝑆 and a number 𝛼 > 0, such that 𝑓(𝑥) = 𝛼𝑥(𝑠𝑓), 𝑥 ∈ 𝐸(𝑆). If 𝜆 𝜈 = sup𝑠∈𝑆 𝑒𝜈 (𝑠) the value sup𝜈

𝑓(𝑒𝜈 ) 𝜆𝜈

is finite for each 𝑓 ∈ Δ(𝐸) due to the relation 𝑒𝜈 (𝑠𝑓) 𝑓𝑒𝜈 ) = ≤ 1. 𝜆𝜈 sup 𝑒𝜈 (𝑠) 𝑠∈𝑆

(6) ⇒ (1). According to Theorem 5.8 the functional ‖𝑥‖ = sup𝑓∈Δ(𝐸)

𝑓(|𝑥|) 𝑐𝑓

is a norm on

𝐸 which obviously is monotone. The closedness of any maximal ideal with respect to this norm is a consequence of the continuity of discrete functionals. The latter follows from the estimate 󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨 ≤ 𝑓(|𝑥|) ≤ 𝑐𝑓 ⋅ sup 𝑓(|𝑥|) = 𝑐𝑓 ‖𝑥‖ , 󵄨 󵄨 𝑐𝑓 𝑓∈Δ(𝐸) which is true for any discrete functional 𝑓.

7.2 Vector lattices of type (Σ) and their representations

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The previous theorem established the existence of qualified e-representations for a vector lattice 𝐸. The constructive approach of the theorem will unfortunately be lost if E-representations are considered. The next result shows that for the existence of an E-representation of 𝐸, a condition must be found which guarantees the local compactness of M(𝐸). Theorem 7.26. For a vector lattice of type (Σ) the following statements are equivalent: (1) a completely regular E-representation exists on a topologicalspace 𝑆 which satisfies the condition: a sequence (𝐾𝑛 )𝑛∈ℕ of compact subsets of 𝑆 exists with the properties (a) 𝑆 = ⋃∞ 𝑛=1 𝐾𝑛 ; and (b) if 𝑥 is a function on 𝑆 whose restrictions to 𝐾𝑛 are all continuous, then 𝑥 is continuous on the whole of 𝑆; (2) a completely regular bE-representation exists; (3) a completely regular 𝑐0 E-representation exists. Proof. (1) ⇒ (2). Let (𝐸(𝑆), 𝑖) be a representation of 𝐸 with the properties listed under (1), and let (𝐾𝑗 )𝑗∈ℕ be a sequence of compact subsets of 𝑆 with the properties (a) and (b). The function 𝑎 constructed in the scheme above is continuous on 𝑆 due to (b). For sufficiently large 𝐶 > 0 each function 𝑥 ∈ 𝐸(𝑆) satisfies the inequality |𝑥| ≤ 𝐶 𝑎. The property of an E-representation is also preserved after the division of all functions 𝑥 ∈ 𝐸(𝑆) by the positive continuous (on 𝑆) function 𝑎. Therefore, (𝐸𝑎 (𝑆), 𝑖) with 𝑥 𝐸𝑎 (𝑆) = { : 𝑥 ∈ 𝐸(𝑆)} 𝑎 is a completely regular bE-representation. (2) ⇒ (3). For a completely regular bE-representation (𝐸(𝑆), 𝑖), the topological space 𝑆 is completely regular and the set 𝐸(𝑆) is isomorphic to some set 𝐸0 (𝛽𝑆) ⊂ 𝐶(𝛽𝑆). Analogous to the proof of the implication (4) ⇒ (5) in the previous theorem, after the factorization of 𝛽𝑆 with respect to the relation 𝜉, we obtain the compact space 𝑇 = 𝛽𝑆/𝜉 , and on 𝑇 the set 𝐸(𝑇) of continuous functions. Let 𝑆0 and 𝑞∞ be the images of 𝑆 and the set 𝐹 = {𝑠 ∈ 𝛽𝑆 : 𝑥(𝑠) = 0 for all 𝑥 ∈ 𝐸0 (𝛽𝑆)} under the quotient map respectively. We show that 𝑇 consists only of 𝑆0 and 𝑞∞ . If there were a residue class [𝑡] not belonging to 𝑆0 ∪ {𝑞∞} and if 𝑡 is some representative of the class [𝑡], then 𝑡 ∈ 𝛽𝑆 \ (𝑆 ∪ 𝐹) and is the limit of an ultrafilter U in 𝑆. Since 𝑡 ∈ ̸ 𝐹, and for each function 𝑥 ∈ 𝐸(𝑆) the (finite) limit lim 𝑥(𝑠) exists, the point 𝑡 defines a U

nonzero discrete functional 𝑓𝑡 on 𝐸(𝑆). Because of the E-representation there is a point 𝑠𝑡 ∈ 𝑆, such that for some 𝛼 > 0 for any 𝑥 ∈ 𝐸(𝑆) there holds 𝑓𝑡 (𝑥) = 𝛼 ⋅ 𝑥(𝑠𝑡 ). Due to the ultrafilter U converging to 𝑡 = 𝑠𝑡 , a set 𝐴 ∈ U exists with 𝑠𝑡 ∉ 𝐴. By assumption there is a function 𝑥0 ∈ 𝐸(𝑆) such that 𝑥0 (𝑠𝑡 ) = 1 and 𝑥0 (𝑠) = 0 for 𝑠 ∈ 𝐴. Then 𝛼 = 0 follows and hence 𝑇 = 𝑆0 ∪ {𝑞∞}. The restrictions 𝐸(𝑆0 ) of all functions of 𝐸(𝑇) on 𝑆0 , together with the isomorphism obtained along the proof , yield a completely regular E-representation which is also a 𝑐0 -representation according to the basic scheme.

156 | 7 Representations of vector lattices and their properties (3) ⇒ (1). If (𝐸(𝑆), 𝑖) is a representation of 𝐸 with the properties listed under (3), then by Theorem 7.16 the space 𝑆 is locally compact and 𝜎-compact. Then it is a countable union ⋃∞ 𝑗=1 𝑆𝑗 of compact sets 𝑆𝑗 ⊂ 𝑆, where 𝑆𝑗 ⊂ int(𝑆𝑗+1 ), 𝑗 ∈ ℕ. It is easy to conclude that the conditions (a) and (b) are satisfied. Remarks 7.27. (1) Any locally compact and 𝜎-compact space satisfies the conditions (a) and (b) in the first part of the theorem. (2) Theorem 7.12 implies that the representations considered in Theorem 7.26 are actual representations on the space M, where the latter is locally compact and 𝜎compact.

8 Vector lattices of continuous functions with finite elements 8.1 Vector lattices of continuous functions with many finite functions In this chapter we study both finite functions and finite elements in vector lattices of continuous functions on a locally compact Hausdorff space, where very close relations between both are expected. Although in general both notions are different, a very mild condition ensures that any finite function is a finite element. However, it is hard for a finite element to be a finite function. In Section 8.1 we consider vector lattices 𝐸(𝑄) of continuous functions defined on a locally compact topological space 𝑄. Due to its rich structure, the vector space 𝐶(𝑄) of all continuous functions is extensively investigated in nearly all branches of functional analysis, e. g., in the classics [49, 112]. Our attention mainly focuses on the vector lattice properties of 𝐶(𝑄) and some of its vector sublattices 𝐸(𝑄) which contain finite functions. Of interest are the interactions between vector lattice properties of 𝐸(𝑄) and the corresponding topological properties of the space 𝑄. Finite elements in 𝐸(𝑄) are studied in Section 8.2. Section 8.3 contains an important result which provides conditions under which the spaces 𝑄 and 𝑆 are homeomorphic if the two vector lattices 𝐸(𝑄) and 𝐹(𝑆) of continuous functions are isomorphic. The representation theory of vector lattices with finite elements, i. e., the possibility of representing abstract vector lattices as a vector lattice of (everywhere finitevalued) continuous functions on a locally compact space, where all finite elements are represented as finite functions, is considered in Chapter 9. The following condition is important both for further study of vector lattices 𝐸(𝑄) of continuous functions, and for the representation theory of general vector lattices containing finite elements. Definition 8.1. A vector lattice 𝐸(𝑄) ⊂ 𝐶(𝑄) satisfies the condition (𝛼), if the set K(𝑄) ∩ 𝐸(𝑄) of all finite functions belonging to 𝐸(𝑄) strongly separates the points of 𝑄 in the following sense: for any ordered pair of points 𝑡0 , 𝑡1 ∈ 𝑄 (𝑡0 ≠ 𝑡1 ) a finite function 𝑥 ∈ 𝐸(𝑄) exists with 𝑥(𝑡𝑘 ) = 𝑘 for 𝑘 = 0, 1. The following example is very interesting. Example 8.2. A vector lattice of type (Σ) consisting of continuous functions not containing finite functions but with finite elements strongly separating the points. Consider the set of almost periodic functions 𝑎P(𝑇) on 𝑇 = (−∞, +∞); see [79]. A continuous function 𝑥 on 𝑇 is called almost periodic if for any 𝜀 > 0 a number 𝑙 = 𝑙(𝑥) exists,

158 | 8 Vector lattices of continuous functions with finite elements such that in any interval of length 𝑙 there is at least one number 𝜏 with |𝑥(𝑡 + 𝜏) − 𝑥(𝑡)| < 𝜀,

𝑡 ∈ 𝑇.

Observe that each periodic function is almost periodic and, except for the zero-function, no finite function is almost periodic. In [79] it is established that under the pointwise algebraic operations and pointwise order, the set 𝑎P(𝑇) of almost periodic functions on 𝑇 is a vector lattice. By means of the functions {| sin 2𝑡 | : 𝑛 ∈ ℕ} define the sets 󵄨 󵄨 𝐸𝑛 = {𝑧 ∈ 𝑎P(𝑇) : |𝑧(𝑡)| ≤ 𝑐𝑧 󵄨󵄨󵄨󵄨sin 2𝑡𝑛 󵄨󵄨󵄨󵄨 , for all 𝑡 ∈ 𝑇 and some 𝑐𝑧 }. Then 𝐸̃ = ⋃∞ 𝑛=1 𝐸𝑛 is a vector lattice of type (Σ). In view of sin

𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 = sin ( 𝑘+1 + 𝑘+1 ) = 2 sin ( 𝑘+1 ) cos ( 𝑘+1 ) ≤ 2 sin ( 𝑘+1 ) 2𝑘 2 2 2 2 2

for all 𝑡 ∈ [0, 2𝑘−1 𝜋] as a sequence (𝑒𝑘)𝑘∈ℕ which satisfies the condition (Σ󸀠 ) in 𝐸̃ can be taken, e. g., the functions 󵄨󵄨 𝑡 󵄨󵄨 𝑒𝑘 (𝑡) = 2𝑘 󵄨󵄨󵄨󵄨sin 𝑘−1 󵄨󵄨󵄨󵄨 , 𝑘 ∈ ℕ. 2 󵄨 󵄨 We describe some of the finite elements in 𝐸.̃ Fix a natural number 𝑚 and a real 𝛿 ∈ (0, 𝜋2 ). Then any function 𝜑 ∈ 𝐸̃ which vanishes on each interval of kind (2𝑚 𝑘𝜋 − 𝛿, 2𝑚 𝑘𝜋 + 𝛿),

𝑘 = 0, ±1, ±2, . . .

is a finite element in 𝐸.̃ Indeed, for 𝑥 ∈ 𝐸̃ the inequality 󵄨 󵄨 󵄨 󵄨 (8.1) |𝑥(𝑡)| ∧ 𝑛 󵄨󵄨󵄨𝜑(𝑡)󵄨󵄨󵄨 ≤ 𝑐𝑥 󵄨󵄨󵄨󵄨sin 2𝑡𝑚 󵄨󵄨󵄨󵄨 𝑡 ∈ 𝑇, ∀𝑛 ∈ ℕ 󵄨 󵄨 obviously holds if 𝑡 ∈ (2𝑚 𝑘𝜋 − 𝛿, 2𝑚 𝑘𝜋 + 𝛿). If 󵄨󵄨󵄨𝑡 − 2𝑚 𝑘𝜋󵄨󵄨󵄨 ≥ 𝛿 for all 𝑘 = 0, ±1, ±2, . . . then the inequality also holds if put sup |𝑥(𝑡)| 𝑡∈𝑇 . 𝑐𝑥 = sin 2𝛿𝑚 Consider now the restriction of all functions of 𝐸̃ on the set [1, ∞) and denote this collection by 𝐸. Then 𝐸 remains a vector lattice of type (Σ) and, as we will show, the finite elements of 𝐸 strongly separate the points of [1, ∞). For 𝑡0 , 𝑡1 ∈ [1, ∞) with 𝑡0 ≠ 𝑡1 take a continuous function 𝜑0 with compact support, such that 𝜑0 (𝑡0 ) = 0 and 𝜑0 (𝑡1 ) = 1. It is clear that 𝜑0 (𝑡) = 0 for 𝑡 ≤ 12 may be assumed. Take a natural number 𝑚 such that 𝑡0 , 𝑡1 < 2𝑚 𝜋 − 12 and supp(𝜑0 ) ⊂ [ 12 , 2𝑚 𝜋 − 12 ]. Extend the function 𝜑0 periodically outside the interval [0, 2𝑚 𝜋]. Then the new function 𝜑0̃ vanishes, in particular at each of the intervals (2𝑚 𝑘𝜋 − 12 , 2𝑚 𝑘𝜋 + 12 )

for

𝑘 = 0, ±1, ±2, . . . .

Therefore, as established above, 𝜑0̃ is finite element in 𝐸.̃ Its restriction to [1, ∞) is then the required finite element in the vector lattice 𝐸.

8.1 Vector lattices of continuous functions with many finite functions

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Vector lattices of continuous functions have very favorable properties if the condition (𝛼) is fulfilled. Proposition 8.3. Let 𝐸(𝑄) be a vector lattice which strongly separates the points of 𝑄, and let be 𝑍 an ideal of 𝐸(𝑄) consisting of finite functions. Then the following conditions are equivalent: (1) 𝑍 satisfies the condition (⋆) ; (2) 𝑍 separates the points of 𝑄 strongly. In particular, the vector lattice 𝐸(𝑄) satisfies the condition (𝛼) . Proof. Only the implication (1) ⇒ (2) needs to be proved. Fix an ordered pair of points 𝑡0 , 𝑡1 ∈ 𝑄, 𝑡0 ≠ 𝑡1 , and choose functions 0 < 𝑧 ∈ 𝑍, and 𝑥0 ∈ 𝐸(𝑄) such that 𝑧(𝑡1 ) > 0

and 𝑥0 (𝑡0 ) = 0,

𝑥0 (𝑡1 ) = 1.

󵄨 󵄨 The function 𝑦 = 𝐶𝑧∧󵄨󵄨󵄨𝑥0 󵄨󵄨󵄨 obviously belongs to 𝑍. One has 𝑦(𝑡0) = 0, and for sufficiently large 𝐶 there is 𝑦(𝑡1 ) = 1. Proposition 8.4. Let 𝐸(𝑄) be a vector lattice of continuous functions which satisfies the condition (𝛼). If 𝐾 ⊂ 𝑄 is a compact subset of 𝑄, and 𝑡0 ∈ 𝑄 a point with 𝑡0 ∉ 𝐾, then a positive finite function 𝑥0 ∈ 𝐸(𝑄) exists, such that 𝑥0 (𝑡0 ) = 0 and 𝑥0 (𝑡) ≥ 1 on 𝐾. Proof. Due to condition (𝛼) for each point 𝑠 ∈ 𝐾 there is a finite function 𝑥𝑠 ∈ 𝐸(𝑄) with 𝑥𝑠 (𝑡0 ) = 0 and 𝑥𝑠 (𝑠) = 1. Since 𝐾 is compact there is a finite number of such functions 𝑥𝑠1 , 𝑥𝑠2 , . . . , 𝑥𝑠𝑁 such that 󵄨 󵄨 sup 󵄨󵄨󵄨󵄨𝑥𝑠𝑖 (𝑡)󵄨󵄨󵄨󵄨 ≥ 12 for all 𝑡 ∈ 𝐾. 1≤𝑖≤𝑁 󵄨 󵄨 Now take 2 sup 󵄨󵄨󵄨󵄨𝑥𝑠𝑖 󵄨󵄨󵄨󵄨 for the required function. 1≤𝑖≤𝑁 Corollary 8.5. Let 𝐸(𝑄) be a vector lattice of continuous functions which satisfies the condition (𝛼). Let 𝜑0 be a positive finite function in 𝐸(𝑄), 𝐾0 a compact subset of int (suppM (𝜑0 )), and 𝑎0 > 0 an arbitrary number. If 𝐾 is a compact subset of 𝑄 such that 𝐾 ∩ 𝐾0 = 0, there is a finite function 𝜓 with the properties 𝜓(𝑡) = 𝜑0 (𝑡) for 𝑡 ∈ 𝐾0

and 𝜓(𝑡) ≥ 𝑎0 for 𝑡 ∈ 𝐾 .

Proof. In view of the proposition, there is a finite function 𝑥0 which vanishes on 𝐾0 and takes on values on 𝐾 not less than 𝑎0 . The required function is now 𝜓 = 𝜑0 +𝑥0 . Theorem 8.6. Let 𝐸(𝑄) be a vector lattice of continuous functions which satisfies the condition (⋆), and let 0 < 𝑧0 be a finite function in 𝐸(𝑄). If 𝐸(𝑄) strongly separates the points of the set 𝐺+ = {𝑡 ∈ 𝑄 : 𝑧0 (𝑡) > 0}, then the following statements hold: (1) for each closed subset 𝐹 ⊂ 𝑄 and any point 𝑡0 ∈ 𝐺+ , 𝑡0 ∉ 𝐹, a finite function 𝑥0 ∈ 𝐸(𝑄) exists such that 𝑥0 (𝑡0 ) = 1 and 𝑥0 (𝑡) = 0 for any 𝑡 ∈ 𝐹; (2) if the vector lattice 𝐸(𝑄) is (r)-complete, then it contains together with the function 𝑧 also each function 𝑦, which satisfies the condition supp(𝑦) ⊂ supp(𝑧).

160 | 8 Vector lattices of continuous functions with finite elements Proof. (1) The function 𝑧1 = 𝑧 (𝑡1 ) 𝑧0 is finite and satisfies the condition 𝑧1 (𝑡0 ) = 1. For 0 0 each point 𝑡 in the compact set 𝐾 = 𝐹 ∩ supp(𝑧0 ) there is a function 0 < 𝑥𝑡 ∈ 𝐸(𝑄) with 𝑥𝑡 (𝑡) = 1 and 𝑥𝑡 (𝑡0 ) = 0. Due to the compactness of 𝐾 for some finite number of points 𝑡1 , 𝑡2 , . . . , 𝑡𝑁 , the function 𝑥̃ = 𝑥𝑡1 ∨ 𝑥𝑡2 ∨ ⋅ ⋅ ⋅ ∨ 𝑡𝑠𝑁 is positive on 𝐾 and vanishes at the point 𝑡0 . For sufficiently large 𝐶 > 0 the function (𝑧1 − 𝐶 𝑥)̃ + is the required one. (2) Observe that two arbitrary different points of 𝐺+ possess disjoint open neighborhoods which are subsets of 𝐺+ . According to the first statement of the theorem, there is a finite function in 𝐸(𝑄) which strongly separates two given different points of 𝐺+ and has its support in 𝐺+ . So the set of functions in 𝐸(𝑄) with support in 𝐺+ strongly separates the points of 𝐺+ , and therefore by the Stone–Weierstrass Theorem, the closure of this set with respect to the uniform convergence on 𝑄 consists of all continuous functions on 𝑄, the supports of which lie in supp(𝑧0 ). In view of Proposition 8.4, it is easy to see that any uniformly convergent sequence of functions with supports in supp(𝑧0 ) is an (𝑟)-Cauchy sequence in the vector lattice 𝐸(𝑄). The proof is finished by referring to the (𝑟)-completeness of 𝐸(𝑄). Corollary 8.7. If the vector lattice 𝐸(𝑄) satisfies the condition (𝛼), then 𝐸(𝑄) separates the points and closed subsets of 𝑄 in the following sense: for each closed set 𝐹 ⊂ 𝑄 and any point 𝑡0 ∈ 𝑄 \ 𝐹, a finite function 𝜑 ∈ 𝐸(𝑄) exists such that 𝜑(𝑡0 ) = 1 and 𝜑(𝑡) = 0 for all 𝑡 ∈ 𝐹. Notice that in this case the space 𝑄 is completely regular. Corollary 8.8. Let 𝐸(𝑄) be a (𝑟)-complete vector lattice of continuous functions which satisfies the condition (𝛼). Then 𝐸(𝑄) contains all finite continuous functions on 𝑄. Proof. Let 𝑦 be an arbitrary finite function on 𝑄. For the compact subset supp(𝑦) ⊂ 𝑄 by Proposition 8.4, there is a finite function 𝑥 ∈ 𝐸(𝑄) with 𝑥(𝑡) ≥ 1 for 𝑡 ∈ supp(𝑦). This means supp(𝑦) ⊂ supp(𝑥). Since 𝐸(𝑄) is (𝑟)-complete according to the second statement of the theorem, the function 𝑦 belongs to 𝐸(𝑄). Now we deal with discrete functionals on vector lattices 𝐸(𝑄). As already mentioned in Section 5.2, in vector lattices of continuous functions which satisfy the condition (⋆) the value of the functions at an arbitrary fixed point 𝑡 ∈ 𝑄 provides a discrete functional, denoted by 𝛿𝑡 . In a certain sense we now solve the inverse problem, namely we ask which discrete functionals on 𝐸(𝑄) are of the kind 𝛿𝑡 for some 𝑡 ∈ 𝑄. Theorem 8.9. Let 𝐸(𝑄) be a vector lattice of continuous functions that satisfies the condition (⋆). Then each discrete functional on 𝐸(𝑄) which does not vanish identically on the set of finite functions in 𝐸(𝑄) is proportional to a functional 𝛿𝑡 for some point 𝑡 ∈ 𝑄. If 𝐸(𝑄) strongly separates the points of 𝑄, then the point 𝑡 is uniquely defined. Proof. For the functional 𝑓 ∈ Δ(𝐸(𝑄)), let 𝑥̃ ∈ 𝐸(𝑄) with 𝑓(𝑥)̃ ≠ 0. According to Proposition 5.10, the sets supp(𝑥) ∩ supp(𝑥)̃ for 𝑥 ∈ 𝐸(𝑄) such that 𝑓(𝑥) ≠ 0 form a ̃ system of closed sets with the finite intersection property in the compact set supp(𝑥),

8.1 Vector lattices of continuous functions with many finite functions

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161

and its intersection is therefore nonempty, ⋂

supp(𝑥) = 𝐾𝑓 ≠ 0.

𝑥∈𝐸(𝑄), 𝑓(𝑥)=0̸

We take 𝑡0 ∈ 𝐾𝑓 and show that the functional 𝑓 is a multiple of 𝛿𝑡0 . Assume the contrary, i. e., 𝑓−1 (0) ≠ 𝛿𝑡−1 (0). Then there is a function 0 < 𝑧 ∈ 𝐸(𝑄) with 𝑧(𝑡0 ) = 0 and 0 𝑓(𝑧) > 0. The condition (⋆) now guarantees the existence of a function 𝑦 ∈ 𝐸(𝑄) which is positive at the point 𝑡0 . The functions 𝑧𝑛 = (𝑧 − 1𝑛 𝑦)+ , 𝑛 = 1, 2, . . . vanish in some neighborhood of the point 𝑡0 . This means that the point 𝑡0 cannot belong to supp(𝑧𝑛). Due to the (𝑟)-continuity of the discrete functional 𝑓, we have 𝑓(𝑧𝑛) > 0 if 𝑛 is sufficiently large. Therefore, 𝐾𝑓 ⊄ supp(𝑧𝑛), in contradiction to the inequality 𝑓(𝑧𝑛 ) > 0. So 𝑓−1 (0) = 𝛿𝑡−1 (0) is proved. Consequently, for some 𝜆 0 > 0 the equality 𝑓 = 𝜆 0 𝛿𝑡0 holds. 0 We show the uniqueness of 𝑡0 by assuming the contrary. If 𝑡1 ∈ 𝑄 is a point different from 𝑡0 , and for some 𝜆 1 > 0 one also has 𝑓 = 𝜆 1 𝛿𝑡1 , then since 𝐸(𝑄) strongly separates the points of 𝑄, there is a function 𝑥0 ∈ 𝐸(𝑄) with 𝑥0 (𝑡0 ) = 0 and 𝑥0 (𝑡1 ) ≠ 0, a contradiction. Remark 8.10. (1) If the points 𝑡0 , 𝑡1 belong to 𝐾𝑓 , then 𝑥(𝑡0 ) = 𝜆𝑥(𝑡1 ) for all 𝑥 ∈ 𝐸(𝑄) and some number 𝜆 = 𝜆(𝑡0 , 𝑡1 ) > 0. (2) We complete now the proof of the necessity of Remark 7.24. If a closed maximal ideal does not contain all finite functions in 𝐸(𝑄), then the corresponding discrete functional does not vanish identically on all finite functions in 𝐸(𝑄). So the maximal ideal is of the kind indicated in the remark mentioned . Theorem 8.9 allows us to conclude that for each finite element 𝜑 ∈ 𝐸(𝑄), the sets 𝑄𝜑 = {𝑡 ∈ 𝑄 : 𝜑(𝑡) ≠ 0}, and 𝐺𝜑 = {𝑀 ∈ M(𝐸(𝑄)), 𝜑 ∉ 𝑀} are homeomorphic. We formulate this fact as follows. Corollary 8.11. Let 𝐸(𝑄) be a vector lattice of continuous functions which strongly separates the points of 𝑄, and let 𝜑 ≥ 0 be a finite function in 𝐸(𝑄). Under the standard map 𝜅 : 𝑄 → M(𝐸(𝑄)) the sets 𝑄𝜑 and 𝐺𝜑 are homeomorphic. Proof. We restrict the map 𝜅 on the set 𝑄𝜑 . According to Section 7.1, one has 𝜅(𝑡) = {𝑥 ∈ 𝐸(𝑄) : 𝑥(𝑡) = 0}, where 𝐸(𝑄) is considered to be its own representation. If 𝑡0 ∈ 𝑄𝜑 , i. e., 𝜑(𝑡0 ) ≠ 0, and therefore the element 𝜑 does not belong to 𝜅(𝑡0 ) = 𝑀0 , then 𝑀0 ∈ 𝐺𝜑 . From this it is clear that first of all 𝜅 maps 𝑄𝜑 into 𝐺𝜑 . Let 𝑀 ∈ 𝐺𝜑 . The corresponding discrete functional 𝑓𝑀 is nonzero on the finite function 𝜑 (which belongs to 𝐸(𝑄)), and according to Theorem 8.9 it is proportional to the functional 𝛿𝑡𝑀 for some unique point 𝑡𝑀 ∈ 𝑄, where it is easy to see that 𝑡𝑀 even belongs to the set 𝑄𝜑 . Consequently, the restriction of 𝜅 to 𝑄𝜑 is surjective and, due to the strong separation of 𝑄, also injective. The continuity of this restriction follows from Theorem 7.6, due to the fact that both sets are open: 𝑄𝜑 in 𝑄 and 𝐺𝜑 in M(𝐸(𝑄)). That the restriction is an open mapping is

162 | 8 Vector lattices of continuous functions with finite elements shown analogue to Proposition 7.9 by, taking Corollary 8.7 to Theorem 8.6 into account. Corollary 8.12. The stronger condition (𝛼) also guarantees that the support and the abstract support of a finite function in 𝐸(𝑄) are homeomorphic. This immediately follows from the previous corollary, since, due to Proposition 8.4, for the compact support supp(𝜓) of a finite function 𝜓 ∈ 𝐸(𝑄) there is a finite function 𝜑 ∈ 𝐸(𝑄) with supp(𝜓) ⊂ 𝑄𝜑 . Now we are able to provide some facts for vector lattices of continuous functions on a topological space 𝑄 which will be used in Chapter 10, when representations by means of bases will be considered. In particular, we establish the relations between 𝑄 and M. Theorem 8.13. Let 𝐸(𝑄) be a vector lattice which consists of finite (continuous) functions and satisfies the condition (𝛼). Then the following statements hold: (1) the space 𝑄 is locally compact; (2) the standard map 𝜅 : 𝑄 → M(𝐸(𝑄)) is a homeomorphism; (3) 𝐸(𝑄) contains all finite functions on 𝑄 if and only if the vector lattice 𝐸(𝑄) is (r)complete. Proof. (1) Due to Theorem 7.3, the local compactness of 𝑄 is obvious. (2) The standard map between 𝑄 and M(𝐸(𝑄)) is a homeomorphism due to the assumptions and Corollary 8.11. (3) For an arbitrary (𝑟)-Cauchy sequence (𝑥𝑛)𝑛∈ℕ of (finite) functions in 𝐸(𝑄), there is a positive function 𝑦0 ∈ 𝐸(𝑄) and a sequence of nonnegative numbers (𝜀𝑛 )𝑛∈ℕ con󵄨 󵄨 verging to zero such that for all 𝑝 = 1, 2, . . . there holds the inequality 󵄨󵄨󵄨󵄨𝑥𝑛+𝑝 − 𝑥𝑛 󵄨󵄨󵄨󵄨 ≤ 𝜀𝑛 𝑦0 for all 𝑛 = 1, 2, . . .. It is easy to see that the sequence (𝑥𝑛 )𝑛∈ℕ compactly converges on 𝑄 to a continuous function 𝑧. By passing to the limit for 𝑝 → ∞ in the last inequality 󵄨 󵄨 we get 󵄨󵄨󵄨𝑧 − 𝑥𝑛 󵄨󵄨󵄨 ≤ 𝜀𝑛𝑦0 , which yields 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 |𝑧| ≤ 󵄨󵄨󵄨𝑧 − 𝑥𝑛 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑥𝑛󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝑥𝑛󵄨󵄨󵄨 + 𝜀𝑛 𝑦0 . Therefore the function 𝑧 is also finite and due to the assumptions belongs to 𝐸(𝑄). Thus it is shown that any (𝑟)-Cauchy sequence in 𝐸(𝑄) has an (𝑟)-limes. The inverse statement is exactly the Corollary 8.8.

8.2 Finite elements in vector lattices of continuous functions In this section, let 𝐸(𝑄) be a vector lattice of continuous functions on some locally compact Hausdorff space 𝑄, i. e., 𝐸(𝑄) ⊂ 𝐶(𝑄). Due to the well-known relation Φ1 (𝐶(𝑄)) = K(𝑄) (see comment, p. 19), it is of interest to deal with vector lattices 𝐸(𝑄) ⊊ 𝐶(𝑄).

8.2 Finite elements in vector lattices of continuous functions

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163

The finite functions in 𝐸(𝑄) are K(𝑄) ∩ 𝐸(𝑄). As usual, the finite elements in 𝐸(𝑄) are denoted by Φ1 (𝐸(𝑄)). In general, (i) K(𝑄) ∩ 𝐸(𝑄) ⊈ Φ1 (𝐸(𝑄)) and

(ii) Φ1 (𝐸(𝑄)) ⊈ K(𝑄) ∩ 𝐸(𝑄).

An example for case (i) follows. Example 8.14. The vector lattice 𝐸 of all continuous functions on [0, ∞) vanishing at 0 contains finite functions which are not finite elements. The finite function 𝑡, if 𝑡 ∈ [0, 1] { { 𝜑(𝑡) = { 2 − 𝑡, if 𝑡 ∈ (1, 2] { if 𝑡 ∈ (2, ∞) { 0, belongs to 𝐸 but fails to be a finite element. Indeed, if 𝑧 were an 𝐸-majorant of 𝜑, then |𝑥| ∧ 𝑛𝜑 ≤ 𝑐𝑥 𝑧 for some 𝑐𝑥 > 0 and any 𝑛 ∈ ℕ. The function 𝑥0 (𝑡) = √𝑧(𝑡) also belongs to 𝐸 and so, with some 𝑐0 > 0, one has √𝑧 ∧ 𝑛𝜑 ≤ 𝑐0 𝑧 for all 𝑛 ∈ ℕ. Since 𝜑(𝑡) > 0 on (0, 1), the infimum on the left coincides with the function √𝑧, i. e., (√𝑧 ∧ 𝑛𝜑) (𝑡) = √𝑧(𝑡), and √𝑧(𝑡) ≤ 𝑐0 𝑧(𝑡) on (0, 1). This implies 0 < 𝑐12 ≤ 𝑧(𝑡), which contradicts 𝑧(𝑡) 󳨀→ 0. 0

𝑡→0

An example for case (ii) is provided next. Example 8.15. A vector lattice of continuous functions containing finite elements which are not finite functions; see also Example 8.2. Consider the vector lattice 𝐸 of all continuous functions 𝑥 on [1, ∞) × [1, ∞) such that 𝑛 ∈ ℕ and 𝜆 > 0 exist with the property |𝑥(𝑡, 𝑠)| ≤ 𝜆 𝑡𝑛 for all (𝑡, 𝑠) ∈ [1, ∞) × [1, ∞). The vector lattice 𝐸 is of type (Σ), where the sequence 𝑒𝑛(𝑡, 𝑠) = 𝑡𝑛

for (𝑡, 𝑠) ∈ [1, ∞) × [1, ∞) and 𝑛 ∈ ℕ

󸀠

satisfies the condition (Σ ) in 𝐸. An element 𝑥 belongs to Φ1 (𝐸) if and only if 𝑥(𝑡, 𝑠) = 0 on a set [𝑎, ∞) × [1, ∞), where 1 ≤ 𝑎 = 𝑎(𝑥). It is clear that not all such functions have a compact support. Compare the finite elements found for the current vector lattice with those obtained for 𝐶(𝑄); see p. 19. For further purposes, we mention that the vector lattice 𝐸 is not a vector lattice of slowly growing functions, since the (later introduced) condition (𝐶𝑀) is not satisfied. In contrast to this, the condition (𝜇) (see Formula (9.3) in Section 9.4) is true. Definition 8.16. A vector lattice 𝐸(𝑄) ⊂ 𝐶(𝑄) satisfies the condition (Φ), if any finite element in 𝐸(𝑄) is a finite function. The vector lattice of the previous example and, due to Lemma 6.36, the vector lattice in Example 6.35 do not satisfy the condition (Φ). The condition¹ (⋆) avoids the case (i), i. e., it holds the following: if a vector lattice 𝐸(𝑄) of continuous functions on a locally compact Hausdorff space 𝑄 satisfies the 1 See page 144.

164 | 8 Vector lattices of continuous functions with finite elements condition (⋆), then any finite function of 𝐸(𝑄) is a finite element of the vector lattice 𝐸(𝑄). Indeed, if a function 𝜑 belongs to K(𝑄) ∩𝐸(𝑄), then, due to the condition (⋆), for any point 𝑡 ∈ supp(𝜑) there is a positive function 𝑥𝑡 ∈ 𝐸(𝑄) with 𝑥𝑡 (𝑡) = 1. The compact set 𝑄𝜑 = supp(𝜑) is covered by the open neighborhoods of a finite number of points 𝑡𝑘 , 𝑘 = 1, 2, . . . , 𝑁 𝑈𝑘 = {𝑡 ∈ 𝑄 : 𝑥𝑡𝑘 (𝑡) > 12 }, 𝑘 = 1, . . . , 𝑁. Denote 𝑧 = 𝑥𝑡1 ∨ 𝑥𝑡2 ∨ ⋅ ⋅ ⋅ ∨ 𝑥𝑡𝑁 . Then 𝑧 ∈ 𝐸(𝑄) and 𝑧(𝑡) > then for sufficiently large 𝑛 ∈ ℕ

1 2

for 𝑡 ∈ 𝑄𝜑 . If now 𝑥 ∈ 𝐸+ (𝑄)

𝑥(𝑡), if 𝜑(𝑡) ≠ 0 󵄨 󵄨 (𝑥 ∧ 𝑛 󵄨󵄨󵄨𝜑󵄨󵄨󵄨)(𝑡) = { . 0, if 𝜑(𝑡) = 0 󵄨 󵄨 In both cases (𝑥 ∧ 𝑛 󵄨󵄨󵄨𝜑󵄨󵄨󵄨)(𝑡) ≤ 2 sup𝑠∈𝑄𝜑 𝑥(𝑠) 𝑧(𝑡), which can be written as 𝑥 ∧ 𝑛𝜑 ≤ 𝑐𝑥 𝑧, where 𝑐𝑥 = 2 sup 𝑥(𝑠). So 𝜑 ∈ Φ1 (𝐸(𝑄)). 𝑠∈𝑄𝜑

Obviously, the condition (Φ) avoids the case (ii). In the present section we prepare some facts which will be needed in the next section in order to prove the main result on isomorphisms of vector lattices consisting of continuous functions and satisfying both conditions (𝛼) and (Φ). We first describe a vector lattice of functions which satisfies the condition (Φ). Let 𝑄 be a locally compact topological space and 𝐸 = 𝐸(𝑄) a vector lattice of type (Σ) which consists of continuous functions on 𝑄and satisfies the condition (𝛼). Denote by (𝑒𝜈 )𝜈∈ℕ a sequence in 𝐸 which satisfies the condition (Σ󸀠 ). We shall call such a vector lattice a vector lattice of majorizing functions, and denote it by 𝐶𝑚𝑎𝑗(𝑄), in more detail by 𝐶𝑚𝑎𝑗 (𝑄, (𝑒𝜈)𝜈∈ℕ ). If the sequence (𝑒𝜈 )𝜈∈ℕ possesses the property: (𝐶𝑀)

for each 𝜈 ∈ ℕ there is a number 𝑘𝜈 such that for any 𝜀 > 0 { the inequality 𝑒𝜈(𝑡) ≤ 𝜀 𝑒𝑘𝜈 (𝑡) holds, whenever 𝑡 does not belong to { some compact subset of 𝑄 depending on 𝜈 and 𝜀, {

then 𝐸 is called a vector lattice of slowly (or merely) growing functions, and in the case that 𝐸(𝑄) is an ideal in 𝐶(𝑄), the notation 𝐶𝑀 (𝑄), in more detail 𝐶𝑀 (𝑄, (𝑒𝜈)𝜈∈ℕ ), will be used. The vector lattice 𝐸 considered in Example 8.15 does not satisfy the condition 1 (𝐶𝑀 ). Indeed, if 𝜈 = 1, 𝑘 + 1 ≥ 2 and 𝜀 > 0, then 𝑡 ≤ 𝜀𝑡𝑘+1 holds for 𝑡 ≥ √𝜀 𝑘 . It suffices

to show that there is no rectangle in the plane ℝ2 such that outside of it one would have 𝑒1 (𝑡, 𝑠) ≤ 𝜀𝑒𝑘 (𝑡, 𝑠). Then the last inequality cannot hold in any compact subset of 1 ℝ either. If 𝑅 = [1, 𝑎] × [1, 𝑏] is a rectangle with 𝑎 ≥ √𝜀 𝑘 , and 𝑏 ≥ 1, then the point 1 ( 2 √𝜀 𝑘 , 2𝑏) ∉ 𝑅 but

𝑒1 (

1 1 1 1 , 2𝑏) = 𝑘 ≥ 𝜀 𝑘 = 𝜀𝑒𝑘 ( 𝑘 , 2𝑏), 𝑘 √𝜀 √𝜀 √𝜀 √𝜀 2 2 2 2

which contradicts the inequality formulated in condition (𝐶𝑀 ). In Section 9.4 we introduce an abstract version of general vector lattices satisfying an appropriate condition and study this class in detail.

8.2 Finite elements in vector lattices of continuous functions

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165

Remark 8.17. (1) The provided condition (𝐶𝑀 ) does not depend on the selected sequence (𝑒𝜈 )𝜈∈ℕ which satisfies the condition (Σ󸀠 ) in 𝐸. It is clear that always 𝜈 < 𝑘𝜈 . (2) In [124]–[126] the introduced vector lattices are considered together with their natural locally convex topology of the inductive limes. Here we neglect any linear topology since the linear topological aspect of representations is not discussed. Proposition 8.18. Any vector lattice of slowly growing functions satisfies the condition (Φ). Proof. Let 𝐸 = 𝐸(𝑄) be a vector lattice of slowly growing functions on a (locally compact) topological space 𝑄, and (𝑒𝜈)𝜈∈ℕ a sequence possessing the properties (Σ󸀠 ) and (𝐶𝑀 ). For an arbitrary finite element 𝑥0 ∈ 𝐸 there is a number 𝜈0 , such that for each 𝜈 ∈ ℕ one finds a real 𝑐𝜈 > 0, satisfying the inequality 𝑒𝜈 ∧ 𝑛𝑥0 ≤ 𝑐𝜈 𝑒𝜈0 for all 𝑛 ∈ ℕ, where, without loss of generality, 𝑥0 > 0 may be assumed. If 𝑥0 were not a finite function, then a sequence of points (𝑡𝑛)𝑛∈ℕ exists, leaving any compact subset of 𝑄, and such that 𝑥0 (𝑡𝑛) > 0 for each 𝑛 ∈ ℕ. Assume 𝑥0 ≤ 𝑐𝑒𝜈1 . For simplicity we may take 𝑐 = 1 and 𝜈1 = 𝜈0 + 1. Then for any 𝜀 ∈ (0, 1) a compact subset 𝐾 ⊂ 𝑄 exists with the property 𝑒𝜈0 (𝑡) ≤

𝜀 𝑒 (𝑡) 𝑐𝜈1 𝜈0+1

for 𝑡 ∈ 𝑄 \ 𝐾.

Obviously there is a number 𝑛0 with 𝑡𝑛0 ∉ 𝐾 and such that for sufficiently large 𝑘 the equation (𝑒𝜈1 ∧ 𝑘𝑥0 )(𝑡𝑛0 ) = 𝑒𝜈1 (𝑡𝑛0 ) holds, from which the inequalities 𝑒𝜈1 (𝑡𝑛0 ) = (𝑒𝜈1 ∧ 𝑘𝑥0 )(𝑡𝑛0 ) ≤ 𝑐𝜈1 𝑒𝜈0 (𝑡𝑛0 ) ≤ 𝜀𝑒𝜈0 +1 (𝑡𝑛0 ) = 𝜀𝑒𝜈1 (𝑡𝑛0 ) follow immediately. This is a contradiction, since 𝑒𝜈1 (𝑡𝑛0 ) > 0 and 𝜀 < 1. We are now able to establish a result for vector lattices of continuous functions which is a special case of the already proved Theorem 6.10. The proof is direct and simpler than the one given for that theorem. Proposition 8.19. Let 𝐸 = 𝐸(𝑄) be a vector lattice of continuous functions satisfying the condition (⋆). If 𝜑 is a finite element in 𝐸 and 𝑧 is one of its (positive) majorants, then supp(𝜑) ⊂ 𝐺 = {𝑡 ∈ 𝑄 : 𝑧(𝑡) > 0}. Proof. Take 𝑡 ∈ supp(𝜑). We show that 𝑧(𝑡) > 0. Assume the contrary, i. e., 𝑧(𝑡) = 0. Then, due to condition (⋆), there is a function 𝑥0 ∈ 𝐸 with 𝑥0 (𝑡) ≠ 0 and 󵄨󵄨󵄨𝑥0 󵄨󵄨󵄨 ∧ 𝑛 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 ≤ 𝑐𝑥 𝑧 󵄨 󵄨 󵄨 󵄨 0

166 | 8 Vector lattices of continuous functions with finite elements holds for some 𝑐𝑥0 > 0 and for all 𝑛 ∈ ℕ. Denote by 𝑈 a neighborhood of 𝑡, in which 󵄨 󵄨 the inequality 𝑧(𝑠) < 𝑐1 󵄨󵄨󵄨𝑥0 (𝑠)󵄨󵄨󵄨 (𝑠) holds. Since in 𝑈 there is a point 𝑡1 with 𝜑(𝑡1 ) ≠ 0, 𝑥0

for sufficiently large 𝑛 one has 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨𝑥0 󵄨󵄨 (𝑡1 ) ∧ 𝑛 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 (𝑡1 ) = 󵄨󵄨󵄨𝑥0 󵄨󵄨󵄨 (𝑡1 ), which yields the impossible inequality 󵄨 󵄨 󵄨 󵄨 𝑐𝑥0 𝑧(𝑡1 ) < 󵄨󵄨󵄨𝑥0 󵄨󵄨󵄨 (𝑡1 ) ∧ 𝑛 󵄨󵄨󵄨𝜑󵄨󵄨󵄨 (𝑡1 ) ≤ 𝑐𝑥0 𝑧(𝑡1). Theorem 8.20. Let 𝐸 = 𝐸(𝑄) be a vector lattice of continuous functions on the topological space 𝑄 which satisfies the conditions (𝛼) and (Φ). Then the following statements hold: (1) the space 𝑄 is locally compact; (2) Φ1 (𝐸) is the set of all finite functions belonging to 𝐸, and Φ1 (𝐸) = Φ2 (𝐸); (3) the spaces 𝑄 and MΦ (𝐸) = ⋃{𝐺𝜑 : 𝜑 ∈ Φ1 (𝐸)} are homeomorphic; (4) 𝐸 contains all finite functions on 𝑄 if and only if 𝐸 is (r)-complete. Proof. The proof of statements (1) and (4) are the same as for statements (1) and (3) in Theorem 8.13. (2) The set 𝑍 = K(𝑄) ∩ 𝐸(𝑄) of all finite functions in 𝐸 on its own is a vector lattice of (continuous) finite functions which satisfies the condition (𝛼). The conditions (Φ) and (𝛼), which are supposed to hold for 𝐸, guarantee the equality 𝑍 = Φ1 (𝐸). By means of Propositions 8.3 and 8.19, the condition (𝛼) implies Φ1 (𝐸) = Φ2 (𝐸). (3) Consider the standard map 𝜅 : 𝑄 → M(𝑍). Due to Proposition 7.7, and since (𝛼) holds in 𝑍, the map 𝜅 is injective, and is surjective due to Theorem 8.9. Since any locally compact Hausdorff space is completely regular (see [94, p. 237]), it follows from Proposition 7.9 that 𝜅 is a homeomorphism. Then 𝑍 = Φ2 (𝐸) implies that 𝑄 and M(Φ2 (𝐸)) are also homeomorphic. According to Corollary 6.18, the spaces MΦ (𝐸) and M(Φ2 (𝐸)) are homeomorphic, and so the proof of statement (3) is complete. Remark 8.21. Theorem 8.13 is a special case of the proved result. Indeed, the condition (Φ) obviously holds in any vector lattice which consists of only finite functions. Then from the proof of statement (3) of the previous theorem, it is clear that in the case of 𝐸 = 𝑍 the standard map 𝜅 is a homeomorphism between 𝑄 and M(𝐸). We are now in a position to provide some more properties of the space of maximal ideals for a special class of vector lattices of continuous functions. As will be shown by Example 9.5, the uniform completeness in the next result seems to be an essential condition. Theorem 8.22. Let 𝐸 = 𝐸(𝑄) be an (r)-complete vector lattice of continuous functions on a 𝜎-compact (noncompact) space 𝑄. If 𝐸 is a vector lattice of type (Σ) which satisfies both conditions (𝛼) and (Φ), then the spaces 𝑄 and M(𝐸) are homeomorphic, and M(𝐸) = MΦ (𝐸) holds. Moreover, in 𝐸(𝑄) a countable sufficient set of finite functions exists.

8.3 An isomorphism result for vector lattices of continuous functions

|

167

Proof. Since by the previous theorem the spaces 𝑄 and MΦ are homeomorphic, it is sufficient to show the equality M = MΦ . The latter will hold if we establish the equalities (i)

MΦ = M

and

(ii) MΦ = MΦ .

Denote by 𝑍 the set of all finite functions belonging to 𝐸, i. e., 𝑍 = K(𝑄) ∩ 𝐸(𝑄). Due to the (𝑟)-completeness of 𝐸, the set 𝑍 contains all finite functions on 𝑄; see Corollary 8.8. Hence the equality 𝑍 = Φ1 (𝐸), which holds according to Theorem 8.20 (2), implies that Φ1 (𝐸) is complete in 𝐸, i. e., 𝑥 ⊥ Φ1 (𝐸) implies 𝑥 = 0. Hence according to Corollary 6.19, MΦ is dense in M, which proves (i). In view of the last theorem, one has also Φ1 (𝐸) = Φ2 (𝐸). The statement (ii) will follow from Theorem 6.32 if we show that Φ2 (𝐸) is a vector lattice of type (Σ). This actually holds (see Corollary 6.20), since 𝑄 is 𝜎-compact and homeomorphic to MΦ . Any sequence which satisfies the condition (Σ󸀠 ) in Φ2 (𝐸) is a countable sufficient set of finite functions in 𝐸. Corollary 8.23. The space M is locally compact and 𝜎-compact (see Remark 6.14 (2)). Corollary 8.24. The vector lattice 𝐸 = 𝐸(𝑄) is its own completely regular E-representation. Corollary 8.25. If a vector lattice 𝐸 = 𝐸(𝑄) satisfies the conditions of the theorem, then any discrete functional on 𝐸 is a multiple of the functional 𝛿𝑡 for some 𝑡 ∈ 𝑄. Corollary 8.26. Since the spaces 𝑄 and M(𝐸) are homeomorphic, it follows that for two isomorphic vector lattices 𝐸(𝑄) and 𝐹(𝑇), each of which satisfies the conditions of the theorem, the spaces 𝑄 and 𝑇 are homeomorphic.

8.3 An isomorphism result for vector lattices of continuous functions The statement of the last corollary can be strengthened as follows (see [89, Theorem 2.2]) Theorem 8.27. Let 𝐸(𝑆) and 𝐹(𝑇) be two lattice isomorphic vector lattices of continuous functions on the topological spaces 𝑆 and 𝑇 respectively. If 𝐸(𝑆) and 𝐹(𝑇) both satisfy the conditions (𝛼) and (Φ), then the spaces 𝑆 and 𝑇 are homeomorphic and each isomorphism 𝑖 : 𝐸(𝑆) → 𝐹(𝑇) has the following structure (𝑖𝑥)(𝑡) = 𝑎(𝑡) 𝑥(ℎ(𝑡)),

𝑥 ∈ 𝐸(𝑆), 𝑡 ∈ 𝑇,

where 𝑎 is a positive continuous function on 𝑇, and ℎ is a homeomorphism from 𝑇 onto 𝑆. Proof. According to Theorem 8.20, the spaces 𝑆 and 𝑇 are homeomorphic (both are also homeomorphic to the common space MΦ of the isomorphic vector lattices). Let ℎ

168 | 8 Vector lattices of continuous functions with finite elements be a fixed homeomorphic mapping from 𝑇 onto 𝑆, 𝑡0 ∈ 𝑇 arbitrary point, and 𝑠0 = ℎ(𝑡0 ). Obviously one has 𝑖(𝛿𝑠−1 (0)) = 𝛿𝑡−1 (0), and hence for 𝑥 ∈ 𝐸(𝑆) the two equations 0 0 𝑥(𝑠0 ) = 0 and

(8.2)

(𝑖𝑥)(𝑡0) = 0

are equivalent. We show that for arbitrary 𝑥1 , 𝑥2 ∈ 𝐸(𝑆), the equation (𝑖𝑥1 )(𝑡0 ) (𝑖𝑥2 )(𝑡0 ) = 𝑥1 (𝑠0 ) 𝑥2 (𝑠0 )

(8.3)

holds, provided 𝑥1 (𝑠0 ) ≠ 0, 𝑥2 (𝑠0 ) ≠ 0. Indeed, the function 𝑧=

𝑥2 𝑥1 − 𝑥2 (𝑠0 ) 𝑥1 (𝑠0 )

belongs to 𝐸(𝑆) and vanishes at the point 𝑠0 . Hence (8.2) implies (𝑖𝑧)(𝑡0) = 0. The computation of the value of the functional 𝛿𝑡0 on both sides of the equation 𝑖𝑥2 𝑖𝑥1 = + 𝑖𝑧 𝑥2 (𝑠0 ) 𝑥1 (𝑠0 ) yields (8.3). The latter now allows the definition of a positive² function 𝑎 at the point 𝑡0 by (𝑖𝑥1 )(𝑡0 ) (𝑖𝑥2 )(𝑡0 ) 𝑎(𝑡0 ) = = . 𝑥1 (𝑠0 ) 𝑥2 (𝑠0 ) The function 𝑎 defined this way on 𝑇 turns out to be continuous, since (8.3) holds at any point of some neighborhood of the point 𝑡0 , where both functions 𝑥𝑘 , 𝑘 = 1, 2 satisfy the condition 𝑥𝑘 (𝑠0 ) = (𝑖 𝑥𝑘 )(𝑡0) ≠ 0. The following examples show that vector lattices of slowly growing functions defined on one and the same topological space are not necessarily isomorphic, even if they all are (𝑟)-complete. Consider on 𝑇 = [1, +∞) the following vector lattices of slowly growing functions 𝐸1

=

𝐶𝑀 (𝑇, (𝑡𝜈)𝜈∈ℕ );

𝐸2

=

𝐶𝑀 (𝑇, (𝑒 𝑡 )𝜈∈ℕ );

𝐸3

=

𝐶𝑀 (𝑇, (𝑒 𝑡

𝜈

1 1− 𝜈

)𝜈∈ℕ ).

By means of Theorem 8.27 we establish that these vector lattices are mutually nonisomorphic. We will show that only for 𝐸1 and 𝐸2 . The other cases follow analogously. Assume the contrary, that 𝐸1 and 𝐸2 are isomorphic. As in any vector lattice of slowly

2 If 𝑥1 (𝑠0 ) > 0, then 𝑥1 (𝑠0 ) = 𝑥+1 (𝑠0 ), i. e., at the point 𝑠0 the function 𝑥0 coincides with the positive function 𝑥+1 . Hence 𝑖(𝑥+1 )(𝑡0) > 0, and so 𝑎(𝑡0 ) = −𝑥−1 (𝑠0 ) and proceed with −𝑥−1 .

(𝑖𝑥1 )(𝑡0 ) 𝑥1 (𝑠0 )

=

(𝑖𝑥+ 1 )(𝑡0 ) 𝑥1 (𝑠0 )

> 0. If 𝑥1 (𝑠0 ) < 0, then 𝑥1 (𝑠0 ) =

8.3 An isomorphism result for vector lattices of continuous functions

| 169

growing functions in both vector lattices, the condition (𝛼) is satisfied by definition and the condition (Φ) holds due to Proposition 8.18. Therefore Theorem 8.27 guarantees that the isomorphism 𝑖 : 𝐸2 → 𝐸1 has the form 𝑥 ∈ 𝐸2 ,

(𝑖𝑥)(𝑡) = 𝑎(𝑡) 𝑥(ℎ(𝑡)),

where 𝑎 is a positive continuous function on 𝑇, and ℎ a homeomorphic mapping from 𝑇 = [1, +∞) onto 𝑇, i. e., ℎ is a continuous strongly increasing function on 𝑇 with ℎ(1) = 1 and lim ℎ(𝑡) = +∞. Since 𝑖 is an isomorphism, the functions 𝑡→∞

𝜈

𝑖 (𝑒𝑡 ) (𝑡) = 𝑎(𝑡) 𝑒 ℎ

𝜈

(𝑡)

,

𝜈∈ℕ

provide a sequence satisfying the condition (Σ󸀠) in 𝐸1 . Consequently, positive numbers 𝐴, 𝐵, 𝐶 exist and indices 𝑚, 𝑝, 𝑞 with 𝑚 < 𝑞 such that 𝐶 𝑡 ≤ 𝑎(𝑡) 𝑒 ℎ

𝑚

(𝑡)

≤ 𝐴 𝑡𝑝

and 𝑡𝑝+2 ≤ 𝐵 𝑎(𝑡) 𝑒 ℎ

𝑞

(𝑡)

(8.4)

.

𝑞+1

We now show that the function 𝑎(𝑡) 𝑒 ℎ (𝑡) increases faster than any polynomial, and therefore cannot belong to the vector lattice 𝐸1 . As a consequence we obtain a contradiction. It follows from (8.4) that 𝑎(𝑡) ≤ 𝐴 𝑡𝑝 𝑒−ℎ

𝑚

(𝑡)

,

𝑎(𝑡) ≥ 𝐶 𝑡 𝑒−ℎ

We estimate the function 𝑎(𝑡)𝑒ℎ 𝑎(𝑡)𝑒

ℎ 𝑞+1(𝑡)

≥ 𝐶𝑡𝑒

𝑞+1

(𝑡)

−ℎ 𝑚(𝑡)

= 𝐶 𝑡(

(𝑡)

and 𝑒 ℎ

𝑞

(𝑡)

≥

1 𝑡𝑝+2 𝑡2 ℎ 𝑚(𝑡) . ≥ 𝑒 𝐵 𝑎(𝑡) 𝐴𝐵

from below

(𝑒

𝑡2 ) 𝐴𝐵

𝑚

ℎ 𝑞(𝑡) ℎ(𝑡)

)

ℎ(𝑡)

𝑒ℎ

𝑚+1

≥ 𝐶𝑡𝑒

(𝑡)−ℎ 𝑚(𝑡)

−ℎ 𝑚 (𝑡)

𝑡2 ( ) 𝐴𝐵

≥ 𝐶 𝑡(

𝑡2 ) 𝐴𝐵

ℎ(𝑡)

𝑒ℎ

𝑚+1

(𝑡)

ℎ(𝑡)

.

For further estimations let 𝑡 be sufficiently large, say, such that 𝐶 𝑡 ≥ 1 and hold. Then 𝑎(𝑡)𝑒 ℎ

𝑞+1

(𝑡)

≥ 𝐶 𝑡(

𝑡 ≥ 1 𝐴𝐵

𝑡2 ℎ(𝑡) ) ≥ 𝑡 ℎ(𝑡). 𝐴𝐵

If 𝑁 ∈ ℕ is fixed, then ℎ(𝑡) → +∞ implies 𝑡ℎ(𝑡)−𝑁 → +∞ as well. Then the estimation 𝑎(𝑡)𝑒 ℎ 𝑡𝑁

𝑞+1

(𝑡)

≥ 𝑡 ℎ(𝑡)−𝑁

𝑞+1

shows that the function 𝑎(𝑡)𝑒 ℎ (𝑡) increases faster than any polynomial 𝑡𝑁 for 𝑁 ∈ ℕ. It is easy to see that the vector lattices of slowly growing functions on 𝑇 = [1, +∞) 𝐶𝑀 (𝑇, (𝑡𝜈 )𝜈∈ℕ )

and

𝐶𝑀 (𝑇, (𝑎(𝑡)ℎ𝜈(𝑡))𝜈∈ℕ )

are isomorphic, where ℎ is a homeomorphic mapping from 𝑇 onto 𝑇, and 𝑎 an arbitrary positive continuous function.

170 | 8 Vector lattices of continuous functions with finite elements Observe that the reason for two vector lattices of slowly growing functions not to be isomorphic is not only a consequence of the fact that both sequences, satisfying the condition (Σ󸀠 ) in these vector lattices, increase in quite different ways, but should be seen more in the fact that these sequences increase in an extremely uncoordinated fashion.

9 Representations of vector lattices by means of continuous functions In the first paragraph we consider representations of vector lattices by means of (finite valued) real continuous functions which contain many finite functions, and then address the question when does a vector lattice 𝐸 possess a representation (𝐸(𝑄), 𝑖), such that the vector lattice 𝐸(𝑄) satisfies the conditions (𝛼) and (Φ). The answer will be given in Section 9.2 for vector lattices of type (Σ), in Section 9.3 for 𝐿𝐹-vector lattices, and in Section 9.4 for vector lattices of type (𝐶𝑀 ). Representations of vector lattices by continuous functions have been studied by a large number of mathematicians. We supplement the list of papers mentioned at the beginning of Chapter 5 by [47, 63, 71, 89, 92, 108, 115, 126, 127], and [134], being quite far from providing a comprehensive overview of the relevant literature.

9.1 Representations which contain finite functions If an Archimedean vector lattice contains nontrivial finite elements and is represented as a vector lattice of continuous (everywhere finite) functions on a locally compact space, then it is very natural to expect, at least under certain conditions, that the finite elements are isomorphically represented as finite functions. We introduce this kind of representation in the following way. Definition 9.1. A (⋆)-representation (𝐸(𝑄), 𝑖) of a vector lattice 𝐸 is called a Φrepresentation, if in 𝐸(𝑄) each finite element is a finite function, i. e., if for at least one isomorphic map 𝑖 : 𝐸 → 𝐸(𝑄), the image of any finite element of 𝐸 is a finite function on 𝑄. A representation is called an 𝛼-representation if the finite functions belonging to 𝐸(𝑄) strongly separate the points of 𝑄. A Φ-representation is called a Φ𝛼-representation if it is also an 𝛼-representation. For a Φ𝛼-representation, the vector lattice 𝐸(𝑄) satisfies the conditions (Φ) and (𝛼). Remark 9.2. (1) For a Φ-representation it is clear that the image of each finite element 𝜑 ∈ 𝐸 is a finite function on 𝑄 under any isomorphism between 𝐸 and 𝐸(𝑄). (2) Due to Theorem 8.20, the topological space 𝑄 is locally compact for any Φ𝛼representation of a vector lattice 𝐸. (3) According to Theorem 8.20, for a Φ𝛼-representation of 𝐸, the topological spaces 𝑆 and MΦ (𝐸) are homeomorphic. (4) Remark (3) immediately implies the following fact: if (𝐸(𝑆), 𝑖) and (𝐸(𝑇), 𝑗) are two Φ𝛼-representations of a vector lattice 𝐸, then the topological spaces 𝑆 and 𝑇 are homeomorphic (see also Theorem 8.27).

172 | 9 Representations of vector lattices by means of continuous functions (5) By Corollary 8.7, each Φ𝛼-representation is completely regular. (6) If an (𝑟)-complete vector lattice of type (Σ) possesses a Φ𝛼-representation on a 𝜎compact space 𝑄, then there also exists a completely regular 𝑐0 𝐸-representation on 𝑄. Indeed, if (𝐸(𝑄), 𝑖) is a Φ𝛼-representation of an (𝑟)-complete vector lattice 𝐸 of type (Σ) on the 𝜎-compact space 𝑄, then in view of Corollary 8.24, the representation (𝐸(𝑄), 𝑖) is a completely regular E-representation. Since 𝑄 is also locally compact, it satisfies conditions (a) and (b) of Theorem 7.26 (1). Hence a completely regular 𝑐0 E-representation for 𝐸 exists. Proposition 9.3. Any completely regular E-representation of a vector lattice is a Φ-representation. Proof. According to Proposition 7.9, in case of a completely regular E-representation (𝐸(𝑄), 𝑖) of the vector lattice 𝐸, the spaces 𝑄 and M(𝐸) are homeomorphic. Hence the statement follows from Theorem 6.10. From the proof it is clear that any representation of a vector lattice 𝐸 on M(𝐸) is a Φ-representation. Theorem 9.4. Any E-representation of a vector lattice, where the ideal of the finite functions satisfies the condition (⋆), is a completely regular Φ𝛼-representation. Proof. Let (𝐸(𝑄), 𝑖) be an E-representation of 𝐸. Due to Proposition 8.3, the vector lattice 𝐸(𝑄) satisfies the condition (𝛼). The complete regularity of the representation follows from Corollary 8.7. Hence, in order to complete the proof it remains to refer to the previous proposition. The inverse statement of this theorem is not true. The next example shows that a vector lattice of type (Σ) exists which possesses a Φ𝛼-representation that is not an E-representation. Example 9.5. A vector lattice of type (Σ) exists consisting of continuous functions on a 𝜎-compact space which satisfies the conditions (Φ) and (𝛼). However, M ≠ MΦ . At the same time, this example shows that the uniform completeness is essential for the result which has been proved in Theorem 8.22 and for Theorem 9.12. For the construction of the actual example we use the vector lattice 𝐹 of Example 6.37 (see p. 141), where 𝐹 is considered a vector lattice 𝐹(𝑇) of continuous functions on the 𝜎-compact space 𝑇 = ℕ × ℕ. 𝐹(𝑇) is of type (Σ) but is not (𝑟)-complete, and from the description of its finite elements in Lemma 6.38, it is easy to see that the condition (Φ) is satisfied. The condition (𝛼) holds trivially. The existence of a discrete functional which vanishes on all finite elements of 𝐹 was indicated in the example mentioned, hence M ≠ MΦ . According to Theorem 8.20 (3), the spaces 𝑇 and MΦ are homeomorphic. Therefore 𝐹(𝑇) as its own representation of 𝐹 turns out to be a Φ𝛼-representation but not an E-representation.

9.1 Representations which contain finite functions

| 173

Next we will demonstrate an application of the general scheme for constructing representations of vector lattices which was exposed in Section 7.2 (and applied in the proofs of Theorems 7.21 and 7.25). The vector lattices now under inspection are vector lattices of infinite matrices and turn out to be similar to those already considered in Examples 6.35 and 6.37. They might also be considered a supplement to the previous example, insofar as the representations are now built up step by step. Remember that the set 𝑇 = ℕ × ℕ is equipped with discrete topology. It is convenient to name some further subsets of 𝑇 = ℕ × ℕ, where for completeness we include the notion of a 𝜑-set. – A subset of 𝑇 is called a 𝜑-set if its intersection with each row is empty or finite; – an infinite subset of 𝑇 is called a 𝛾-set if it is contained in a finite number of rows; – a subset of 𝑇 is called a 𝜓-set if it can be written as 𝑇 \ 𝐴, where 𝐴 is a union of a 𝛾-set and a 𝜑-set. In accordance with the above classification of subsets of 𝑇, we now describe the ultrafilters in 𝑇. Our aim is twofold: – First, to identify the points of the Stone–Čech compactification 𝛽𝑇 of the (discrete) space 𝑇 with the (usual) ultrafilters in 𝑇 (see [49, § 4.1 and Chap. 6]), i. e., by joining to 𝑇 one new point for each nontrivial ultrafilter. – Second, starting from 𝛽𝑇 to directly construct representations for some vector lattices of infinite matrices. In general, in order to do the first step, one has to take all 𝑧-ultrafilters built up from the zero-sets in the space 𝑇 (see [49, § 2.2]); however, for a discrete space every set is zero-set. We frequently use the terminology introduced in the beginning of Section 6.1. Let U be an ultrafilter in 𝑇. As usual, U will be called fixed or trivial if it is generated by some point 𝑡 ∈ 𝑇, i. e., U consists of all subsets of 𝑇 which contain the point 𝑡. Further on, a nontrivial ultrafilter U will be called a – 𝛾-ultrafilter if it contains some 𝛾-set; – 𝜑-ultrafilter if it contains some 𝜑-set; – 𝜓-ultrafilter if it contains all 𝜓-sets. The specified ultrafilters in 𝑇 will be denoted by 𝑇, 𝑇𝛾 , 𝑇𝜑 , 𝑇𝜓 respectively. We claim that there are no other ultrafilters in 𝑇, which means that each ultrafilter in 𝑇 belongs to one of the four classes. Indeed, let U be an arbitrary ultrafilter in 𝑇. If U is not a 𝜓-ultrafilter, then a 𝜓-set 𝐴 ⊂ 𝑇 exists such that 𝑇 \ 𝐴 ∈ U. The set 𝑇 \ 𝐴 is the union of some 𝛾-set 𝐵 and some 𝜑-set 𝐶. Due to 𝑇 \ 𝐴 ∈ U, one of the sets 𝐵, 𝐶 must belong to U (see [31, § I.6.4]), and depending on which one that is, the ultrafilter U is either a 𝛾-ultrafilter or a 𝜑- one. It is now established that after identifying all points of the Stone–Čech compactification 𝛽𝑇 with the ultrafilters of the space 𝑇, one has 𝛽𝑇 = 𝑇 ∪ 𝑇𝛾 ∪ 𝑇𝜑 ∪ 𝑇𝜓 .

174 | 9 Representations of vector lattices by means of continuous functions We now consider two examples of vector lattices of infinite matrices and directly construct representations for them with several particular properties. Both examples essentially are due to B. M. Makarov. In both examples the matrices 𝑒𝜈 , 𝜈 ∈ ℕ are used, which were defined in Formula (6.17). Example 9.6. Let 𝐸 consist of all matrices 𝑥 = (𝑥𝑖𝑗 )𝑖,𝑗∈ℕ which satisfy the conditions (i) numbers 𝜆 and 𝜈 exist such that 󵄨 󵄨 (𝜈) |𝑥| ≤ 𝜆𝑒𝜈 , i. e., 󵄨󵄨󵄨󵄨𝑥𝑖𝑗 󵄨󵄨󵄨󵄨 ≤ 𝜆𝑒𝑖𝑗 for 𝑖, 𝑗 ∈ ℕ; (ii) each matrix is constant at some 𝜓-set. The matrices are considered to be continuous functions at the discrete topological space 𝑇. The finite elements of 𝐸 are the same as those described in Lemma 6.36 for the vector lattice considered in Example 6.35, i. e., their support is a 𝜑-set. All functions of 𝐸 possess finite limits along ultrafilters which do not belong to 𝑇𝛾 . Therefore, according to the Stone–Čech compactification Theorem ([49, Theorem 6.5]), they can be continuously extended to 𝑇 ∪ 𝑇𝜑 ∪ 𝑇𝜓 . For the extension of 𝑥 ∈ 𝐸 to the indicated set, take the restriction on 𝑇 ∪ 𝑇𝜑 ∪ 𝑇𝜓 of the Stone–Čech extension¹ of the function 𝑒𝑥 , 𝜈 where 𝜈 is the number from (i). Since for U1 , U2 ∈ 𝑇𝜓 one has lim 𝑥 = lim 𝑥 U1

U2

for all

𝑥 ∈ 𝐸,

we identify the points of 𝑇𝜓 and get the quotient space 𝑆 = 𝑇 ∪ 𝑇𝜑 ∪ {𝑡∗ }, where 𝑡∗ is the class corresponding to the set 𝑇𝜓 . If we denote the (throughout) extension of 𝑥 ∈ 𝐸 to 𝑆 by 𝑥̃ we get a (⋆)-representation 𝐸(𝑆) of the vector lattice 𝐸. We show that 𝐸(𝑆) also satisfies the condition (Φ). If 𝑥 ∈ Φ1 (𝐸), then supp(𝑥)̃ is a compact set in 𝛽𝑇 which is contained in 𝑇 ∪ 𝑇𝜑 . Therefore, supp(𝑥)̃ is a compact set ̃ ∗) = 0 for any finite element 𝑥 ∈ 𝐸, hence condition (𝛼) is not in 𝑆. Observe that 𝑥(𝑡 satisfied. We now prove that any discrete functional on 𝐸(𝑆) is proportional to a functional of the kind 𝛿𝑠 for some 𝑠 ∈ 𝑆, hence there is a bijection between 𝑆 and M(𝐸). Let 𝑓 be an arbitrary discrete functional on 𝐸. There are two cases: (a) The functional 𝑓 does not vanish identically on the set Φ1 (𝐸(𝑆)). Then, in view of Theorem 8.9, the functional 𝑓 is a multiple of some functional 𝛿𝑠 , where 𝑠 ≠ 𝑡∗ . (b) The functional 𝑓 vanishes at each finite element of 𝐸(𝑆). We now establish the existence of a number 𝐴 > 0 such that 𝑓(𝑥) = 𝐴 𝛿𝑡∗ (𝑥)̃ for all 𝑥 ∈ 𝐸. For this purpose we show that 𝑓 vanishes on each 𝑥 ∈ 𝐸, the support of which is a 𝛾-set.

1 To the whole 𝛽𝑇 which is possible due to the boundedness on 𝑇 of the functions

𝑥 . 𝑒𝜈

9.1 Representations which contain finite functions

|

175

Denote by 𝑎𝜈 the element in 𝐸 such that its first 𝜈 rows coincide with the first 𝜈 rows of the element 𝑒𝜈 , and the remaining rows are zero-rows. Since each element with a 𝛾-set as its support is majorized by some element 𝑐𝑎𝜈 , it is sufficient to show that 𝑓 vanishes on all elements 𝑎𝜈 (𝜈 = 1, 2, . . .). Fix some 𝜈 ∈ ℕ. For each natural 𝑘 the element 𝑎𝜈 can be represented as s sum 𝑎𝜈 = 𝑎𝜈(𝑘) + 𝑏𝑘 , where the first 𝑘 columns of the element 𝑎𝜈(𝑘) are zero-columns, but the remainder coincide with the corresponding columns of 𝑎𝜈 , and the first 𝑘 columns of 𝑏𝑘 coincide with those of 𝑎𝜈 , but the remainder are zero-columns. For example, in the case 𝑘 = 3 the elements 𝑎𝜈(3) and 𝑏3 are the following matrices: 0 0 0 4𝜈 5𝜈 . . . 0 0 0 4𝜈 5𝜈 . . . ( ....................... 𝑎𝜈(3) = ( 0 0 0 4𝜈 5𝜈 . . . 0 0 0 0 0 ... .. .. .. . ... ( . . 𝜈 𝜈 3 0 0 ... 1 2 1 2𝜈 3𝜈 0 0 . . . ( ....................... 𝑏3 = ( 1 2𝜈 3𝜈 0 0 . . . 0 0 0 0 0 ... .. .. .. . ... . ( .

𝑗𝜈 𝑗𝜈 𝑗𝜈 0

0 0 0 0

... ... ) ... ), ... .. . ) ... ... ) ... ). ... .. . )

Obviously, for any 𝑘 the element 𝑏𝑘 is finite, which implies 𝑓(𝑏𝑘 ) = 0. Since for the (𝑘) elements 𝑎𝜈 = 𝑎𝜈(𝑘) + 𝑏𝑘 and 𝑎𝜈+1 = 𝑎𝜈+1 + 𝑏𝑘󸀠 (𝑘) 𝑎𝜈+1 ≥ 𝑘 𝑎𝜈(𝑘)

holds, one has the estimation (𝑘) (𝑘) ) + 𝑓(𝑏𝑘󸀠 ) = 𝑓(𝑎𝜈+1 ) ≥ 𝑘𝑓(𝑎𝜈(𝑘) ) = 𝑘𝑓(𝑎𝜈 ), 𝑓(𝑎𝜈+1 ) = 𝑓(𝑎𝜈+1

which can be true only if 𝑓(𝑎𝜈 ) = 0. So it is proved that 𝑓 vanishes on all elements of 𝐸 with a 𝛾-set as the support². We are now able to complete the proof that 𝑓 is proportional to 𝛿𝑡∗ . It is clear that there is a number 𝑛0 with 𝑓(𝑒𝑛0 ) ≠ 0. For an arbitrary element 𝑥 ∈ 𝐸 there are a real 𝑐 > 0 and a natural 𝑁 such that |𝑥| ≤ 𝑐 𝑒𝑁 , where 𝑁 > 𝑛0 may be assumed. Denote the following elements – 𝑥𝛾 , 𝑒𝛾𝑛0 – the element, where the first 𝑁 rows coincide with the first 𝑁 rows of the element 𝑥, 𝑒𝑛0 and the remaining rows are zero-rows, respectively;

2 Notice that we have used only the fact that 𝑓 vanishes on all finite elements which are different from zero only at a finite number of points of 𝑇.

176 | 9 Representations of vector lattices by means of continuous functions –

–

𝑥𝜓 , 𝑒𝜓𝑛0 – the element, which coincides with the element 𝑥, 𝑒𝑛0 on the intersection of the two sets, where the matrices 𝑥 and 𝑒𝑛0 are constant and are zero outside this intersection, respectively; 𝑥𝜑 = 𝑥 − 𝑥𝛾 − 𝑥𝜓 and 𝑒𝜑𝑛0 = 𝑒𝑛0 − 𝑒𝛾𝑛0 − 𝑒𝜓𝑛0 .

Then we can write 𝑒𝑛0 = 𝑒𝛾𝑛0 + 𝑒𝜑𝑛0 + 𝑒𝜓𝑛0 and 𝑥 = 𝑥𝛾 + 𝑥𝜑 + 𝑥𝜓 , where 𝑥𝜓 = 𝜆 𝑒𝜓𝑛0 for some number 𝜆 > 0, depending upon 𝑥. We have 𝑥𝜓 = 𝜆𝑒𝜓𝑛0 = 𝜆(𝑒𝑛0 − 𝑒𝛾𝑛0 − 𝑒𝜑𝑛0 ) and so, 𝑥 = 𝜆𝑒𝑛0 + (𝑥𝛾 − 𝜆𝑒𝛾𝑛0 ) + (𝑥𝜑 − 𝜆𝑒𝜑𝑛0 ). Due to the property of the functional 𝑓, one has 𝑓(𝑥𝛾 − 𝜆𝑒𝛾𝑛0 ) = 𝑓(𝑥𝜑 − 𝜆𝑒𝜑𝑛0 ) = 0. This implies 𝑓(𝑥) = 𝜆 𝑓(𝑒𝑛0 ) = 𝐴 𝜆. It is clear that the number 𝜆 is the limit of the function ̃ ∗ ). 𝑥 along the ultrafilter U∗ (corresponding to the point 𝑡∗ ), i. e., 𝜆 = lim 𝑥(𝑡) = 𝑥(𝑡 ∗ U

Consequently, 𝑓(𝑥) = 𝐴 𝛿𝑡∗ (𝑥)̃ for all 𝑥 ∈ 𝐸. Summing up the properties of the representation 𝐸(𝑆) of the primary vector lattice 𝐸, we have (1) 𝐸(𝑆) is an E-representation, i. e., the standard map 𝜅 : 𝑆 → M(𝐸) is bijective; (2) the set Φ1 (𝐸) of all finite elements is not sufficient in 𝐸, since the limit of any finite element along the ultrafilter U∗ is equal to 0; (3) 𝐸(𝑆) satisfies the condition (Φ) and so it is a Φ-representation of 𝐸; (4) the condition (𝛼) is not satisfied, since the finite functions of 𝐸(𝑆) all vanish at the point 𝑡∗ ; (4) if consider on 𝑆 the topology 𝜏ℎ𝑘 , then the point 𝑡∗ does not possess any 𝜏ℎ𝑘 -compact neighborhood (see Theorem 6.13 (4)). If we denote 𝑆0 = 𝑇 ∪ 𝑇𝜑 and the restrictions of the functions from 𝐸(𝑆) on the set 𝑆0 by 𝐸(𝑆0 ), then we get another representation of the vector lattice 𝐸, which still meets the condition (Φ). However, the loss of 𝑡∗ results in a representation which can not be an E-representation. Now the condition (𝛼) is also satisfied. Indeed, take 𝑡1 , 𝑡2 ∈ 𝑇𝜑 , 𝑡1 ≠ 𝑡2 and let U1 , U2 denote the corresponding 𝜑-ultrafilters. Since U1 ≠ U2 there is a set 𝑈 ∈ U1 which does not belong to U2 . The 𝜑-ultrafilter U1 contains some 𝜑-set 𝐴, so that 𝐵 = 𝐴 ∩ 𝑈 ∈ U1 and 𝐵 is a 𝜑-set. Since 𝑈 ∉ U2 the set 𝐵 does not belong to U2 either. Consequently, 𝑇 \ 𝐵 ∈ U2 . The characteristic function 𝜒𝐵̃ of 𝐵 is a finite element in 𝐸(𝑆) with the property 𝜒𝐵̃ (𝑡1 ) = 1 and 𝜒𝐵̃ (𝑡2) = 0, which shows that the condition (𝛼) is satisfied. We conclude that 𝐸(𝑆0 ) is a Φ𝛼representation of a vector lattice of type (Σ), not being an E-representation (cf. Example 9.5). We provide now the Example 9.7. An example of a vector lattice of type (Σ) which possesses a noncompletely regular (⋆) Φ 𝑐0 E-representation on a locally compact 𝜎-compact space which is not homeomorphic to M.

9.2 The existence of Φ𝛼-representations for vector lattices of type (Σ)

|

177

Let 𝐹 be the vector lattice 𝐹 ⊂ 𝐸 containing all 𝑥 of 𝐸 (in the previous example), which are constant outside of some 𝛾-set. Similar to the vector lattice in Example 6.37, one can show that 𝐹 is a not uniformly complete vector lattice of type (Σ) and its finite elements are the same as described in Lemma 6.38. Consider the set 𝑆1 = 𝑇 ∪ {𝑡F } (𝑇 enriched with an “artificial” point), and extend all elements (understood as continuous functions on 𝑇) of the vector lattice 𝐹 on 𝑆1 by the formula 𝑥(𝑡F ) := lim 𝑥, F

where F is the filter with the basis consisting of the complements to the 𝛾-sets. If the set 𝑆1 is equipped with discrete topology then we get a representation 𝐹(𝑆1 ) of the vector lattice 𝐹 which is obviously a (⋆)-representation. The finite elements in 𝐹(𝑆1 ) are exactly those functions on 𝑆1 which vanish at the point 𝑡F . We show that 𝐹(𝑆1 ) is an E-representation. Let 𝑓 be a discrete functional on 𝐹. If it does not vanish identically on the set Φ1 (𝐹), then, due to Theorem 8.9, it is proportional to a functional 𝛿𝑡 for some 𝑡 ∈ 𝑇. If 𝑓(𝑥) = 0 for all 𝑥 ∈ Φ(𝐹), then analogously, as in the previous example, it can be shown that 𝑓 = 𝐴 𝛿𝑡 . Hence, 𝐹(𝑆1 ) is an E-representation of 𝐹. F In order to force the condition (𝑐0 ) define a function 𝑢 on 𝑆1 by 𝑢(𝑡) = {

1, 𝑒−𝑖𝑗,

𝑡 = 𝑡F . 𝑡 = (𝑖, 𝑗) ∈ 𝑇

and multiply all functions of 𝐹(𝑆1 ) by 𝑢. We thus obtain a new representation 𝐹1 (𝑆1 ) of the vector lattice 𝐹. It can easily be shown that 𝐹1 (𝑆1 ) is a 𝑐0 -representation which satisfies the condition (Φ). This way, 𝐹1 (𝑆1 ) is a (⋆) Φ 𝛼 𝑐0 E-representation of 𝐹. However, the space M(𝐹) is not homeomorphic to 𝑆1 , since the maximal ideal corresponding to the point 𝑡F contains all finite elements of 𝐹 and therefore, by Theorem 6.13 (4), fails to possess a compact neighborhood in M(𝐹), whereas the point 𝑡F belongs to the discrete space 𝑆1 . The latter fact means that the standard map 𝜅 : 𝑆1 → M(𝐹) is not homeomorphic. According to Proposition 7.9, the representation 𝐹1 (𝑆1 ) cannot be completely regular.

9.2 The existence of Φ𝛼-representations for vector lattices of type (Σ) In Section 7.2 the existence of a Riesz norm on a vector lattice of type (Σ) was shown to be an important condition for the vector lattice to have qualified representations. In the present section we continue our investigation of vector lattices of type (Σ). In particular, it is shown that if a vector lattice 𝐸 possesses a countable complete set of finite elements, then a monotone norm exists, and conditions for the existence of a Φ𝛼-representation are finally provided.

178 | 9 Representations of vector lattices by means of continuous functions Theorem 9.8. If a vector lattice 𝐸 of type (Σ) possesses a countable complete collection of finite elements then a Riesz norm exists on 𝐸 and hence (due to Theorem 7.21), a (⋆)𝑐0 representation exists on a locally compact and 𝜎-compact space. Proof. Let (𝑒𝜈)𝜈∈ℕ be a sequence of elements in the vector lattice 𝐸 which satisfies the condition (Σ󸀠 ), and (𝜑𝑛 )𝑛∈ℕ be a countable complete set of finite elements, i. e., if 𝑥 ⊥ 𝜑𝑛 for any 𝑛 ∈ ℕ, then 𝑥 = 0. Fix a majorant 𝑧𝑛 for each 𝜑𝑛 and denote by (𝑝𝑛)𝑛∈ℕ the corresponding system of seminorms in 𝐸, i. e., 󵄨 󵄨 𝑝𝑛 (𝑥) = 𝑝𝜑𝑛 (𝑥) = inf{𝜆 > 0 : |𝑥| ∧ 𝑐 󵄨󵄨󵄨𝜑𝑛 󵄨󵄨󵄨 ≤ 𝜆𝑧𝑛, 𝑐 > 0} (see Section 3.1). Select a sequence of positive numbers (𝜆 𝑛)𝑛∈ℕ such that for all 𝜈 = 1, 2, . . . the relations 𝜆 𝑛 𝑝𝑛(𝑒𝜈 ) 󳨀→ 0 (9.1) 𝑛→∞

hold, and put

∞

‖𝑥‖ = ∑ 𝑛=1

𝜆𝑛 𝑝 (𝑥) for 2𝑛 𝑛

𝑥 ∈ 𝐸.

Since 𝐸 is a vector lattice of type (Σ), the correctness of the definition of ‖𝑥‖ for any 𝑥 ∈ 𝐸 will follow if the definition only for the elements 𝑒𝜈 , 𝜈 ∈ ℕ is shown to be correct. If for each 𝜈 ∈ ℕ, the number 𝑁 is taken sufficiently large (such that in (9.1) 󵄨 󵄨 one has 󵄨󵄨󵄨𝜆 𝑛𝑝𝑛 (𝑒𝜈)󵄨󵄨󵄨 ≤ 1), then one has ∞ 𝑁 ∞ 𝜆 𝜆 1 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑒𝜈 󵄩󵄩 = ∑ 𝑛𝑛 𝑝𝑛(𝑒𝜈 ) ≤ ∑ 𝑛𝑛 𝑝𝑛(𝑒𝜈 ) + ∑ 𝑛 < ∞. 𝑛=1 2 𝑛=1 2 𝑛=𝑁+1 2

The function ‖⋅‖ is a norm, since, due to the completeness of the system (𝜑𝑛)𝑛∈ℕ , the equality ‖𝑥‖ = 0 implies 𝑥 = 0 (see property 5 on p. 22). The other properties of a norm are obviously, in particular, its monotonicity followed by the monotonicity of the seminorms 𝑝𝑛. Before we deal with the main result on Φ𝛼-representations for vector lattices of type (Σ), we first prove two propositions which are important and useful for further purposes in this section. In the next proposition it is shown that for a vector lattice of type (Σ) with a Riesz norm, a representation (𝐸(𝑄), 𝑖) can always be constructed such that a given sequence of finite elements of 𝐸 is represented as a sequence of finite functions on the space 𝑄. Proposition 9.9. Let 𝐸 be a vector lattice of type (Σ) which possesses a Riesz norm. Let (𝜑𝑛 )𝑛∈ℕ be a sequence of finite elements in 𝐸. Then a (⋆)-representation (𝐸(𝑄), 𝑖)) of 𝐸 exists with the following properties: (1) the space 𝑄 is locally compact and 𝜎-compact; (2) 𝐸(𝑄) separates the points of 𝑄; (3) 𝑖𝜑𝑛 is a finite function for any 𝑛 ∈ ℕ; (4) 𝐸(𝑄) strongly separates the points of the set ⋃∞ 𝑛=1 supp(𝑖𝜑𝑛 ).

9.2 The existence of Φ𝛼-representations for vector lattices of type (Σ)

| 179

Proof. For the first part of the proof we apply the scheme exposed before Theorem 7.21 (see p. 151) for constructing representations for vector lattices of type (Σ). This approach is allowed because the existence of a Riesz norm on 𝐸 is postulated. Consider an arbitrary finite element of the given sequence (𝜑𝑛)𝑛∈ℕ , and trace its behavior within the scheme. Denote by (𝑒𝜈 )𝜈∈ℕ a sequence which satisfies the condition (Σ󸀠 ) in 𝐸. In view of the finiteness of each 𝜑𝑛 , there are an index 𝜈𝑛 and a real 𝑐𝜈,𝑛 > 0 such that the inequality 󵄨 󵄨 𝑒𝜈 ∧ 𝑐 󵄨󵄨󵄨𝜑𝑛󵄨󵄨󵄨 ≤ 𝑐𝜈,𝑛 𝑒𝜈𝑛 holds for arbitrary 𝑐 > 0 and all 𝜈 ∈ ℕ. Let (𝐸(𝑆0 ), 𝑖0 ) be a (⋆)𝑐0 -representation of 𝐸 (on a locally compact and 𝜎-compact topological space 𝑆0 ) which exists according to Theorem 7.21. Denote and

𝑖0 (𝑒𝜈) = 𝑒𝜈, 𝑖0 (𝜑𝑛 ) = 𝜑𝑛

𝐹𝑛 = supp(𝜑𝑛)

𝜈, 𝑛 = 1, 2, . . . .

By way of contradiction, one can show that for 𝑠 ∈ 𝐹𝑛 the inequality 𝑒𝜈 (𝑠) ≤ 𝑐𝜈,𝑛 𝑒𝜈𝑛 (𝑠) holds and moreover, on the set 𝐹𝑛 one has 𝑒𝜈𝑛 (𝑠) > 0. Now select such numbers 𝜆 𝜈 that all series ∞

∞

∑ 𝜆 𝜈 𝑐𝜈,𝑛

for 𝑛 ∈ ℕ

and

𝜈=1

∑ 𝜆 𝜈 max 𝑒𝜈 (𝑠) 𝑠∈𝑆0

𝜈=1

simultaneously converge. The function ∞

𝑒(𝑠) = ∑ 𝜆 𝜈 𝑒𝜈 (𝑠) for 𝑠 ∈ 𝑆0

(9.2)

𝜈=1

is continuous on 𝑆0 and positive at any point 𝑠 ∈ 𝑆0 . On the set 𝐹𝑛 one has the estimate 𝜈𝑛

∞

𝑒(𝑠) ≤ ∑ 𝜆 𝜈 𝑒𝜈 (𝑠) + ∑ 𝜆 𝜈 𝑐𝜈,𝑛 𝑒𝜈𝑛 (𝑠) ≤ 𝐶𝑛 𝑒𝜈𝑛 (𝑠), 𝜈=1

𝜈=𝜈𝑛 +1

where 𝐶𝑛 > 0 is an appropriate coefficient. Next divide all functions of 𝐸(𝑆0 ) by the ˘ function 𝑒 and then extend them to the Stone–Cech compactification 𝛽𝑆0 of the space ̃ 0 ), i𝛽 ). Denote 𝑖𝛽 (𝑒𝜈 ) 𝑆0 . So we get a new representation of 𝐸 which is denoted by (𝐸(𝛽𝑆 ̃𝑛 , respectively. Since supp(̃ and 𝑖𝛽 (𝜑𝑛) by 𝑒̃𝜈 and 𝜑 𝜑𝑛 ) is the closure of the set 𝐹𝑛 in 𝛽𝑆0 and on 𝐹𝑛 the inequality 𝑒̃ 𝜈𝑛 (𝑠) =

𝑒𝜈𝑛 (𝑠) 𝑒(𝑠)

≥

𝑒𝜈𝑛 (𝑠) 𝑐𝑛 𝑒𝜈𝑛 (𝑠)

=

1 > 0. 𝑐𝑛

holds. We conclude that supp(𝜑̃𝑛) is a compact set, on which the function ̃ 𝑒𝜈𝑛 is strongly positive. According to Theorem 7.16 (1), the set ̃ 0 )} ̃ = 0 for any 𝑥̃ ∈ 𝐸(𝛽𝑆 𝐹 = {𝑠 ∈ 𝛽𝑆0 : 𝑥(𝑠)

180 | 9 Representations of vector lattices by means of continuous functions is not empty and disjoint to all sets supp(𝜑̃𝑛), since 𝑒̃𝜈𝑛 nowhere vanishes on supp(𝜑̃𝑛 ). One has 𝐹 ∩ supp(𝜑̃𝑛 ) = 0 for all 𝑛 ∈ ℕ. When now passing from 𝛽𝑆0 to the quotient space 𝛽𝑆0 /𝜉 , where 𝜉 is the equivalence relation 𝑡𝜉𝑡󸀠 for 𝑡, 𝑡󸀠 ∈ 𝛽𝑆0 , if 𝑥(𝑡) = 𝑥(𝑡󸀠 ) for ̃ 0 ) (see the remark before Theorem 7.21), then the images of the all functions 𝑥 ∈ 𝐸(𝛽𝑆 ̃ functions from 𝐸(𝛽𝑆0 ) are continuous functions on the compact space 𝛽𝑆0 /𝜉 . Finally, consider the collection 𝐸(𝑄) of all restrictions of thus obtained functions onto the set 𝑄 = 𝛽𝑆0 /𝜉 \ [𝐹], where [𝐹] denotes the factor class of 𝐹 (under 𝜉). We get an isomorphic mapping 𝑖 throughout, from 𝐸 onto 𝐸(𝑄), and thus arrive at our final representation of 𝐸 which will be denoted by (𝐸(𝑄), 𝑖). Obviously, (𝐸(𝑄), 𝑖) is a (⋆)𝑐0 -representation. So, by Corollary 7.19, the first statement of the proposition is proved. The construction shows that the second statement also holds. It remains to show the further properties for the vector lattice 𝐸(𝑄) and for the isomorphism 𝑖. The supports of the functions 𝑖𝜑𝑛 in 𝑄 are the images of the sets supp(𝜑̃𝑛) ⊂ 𝛽𝑆0 under the canonical quotient map from 𝛽𝑆0 \ 𝐹 onto 𝑄, and are therefore also compact. The ̃ 0 ) strongly separate last statement of the proposition means that the functions of 𝐸(𝛽𝑆 ∞ nonequivalent, with respect to 𝜉, points of the set ⋃𝑛=1 supp(𝜑̃𝑛 ). Let 𝑠0 ∈ supp(𝜑̃𝑛0 ) ̃ 0 ), taking on and 𝑠1 ∈ supp(𝜑̃𝑛1 ) be two points for which there is no function in 𝐸(𝛽𝑆 the values 0 at 𝑠0 and 1 at 𝑠1 . Then the two functionals 𝛿𝑠0 and 𝛿𝑠1 considered on the vector lattice possess a common hyperplane and are therefore proportional. Consẽ 0 ) = 𝜆 𝑥(𝑠 ̃ 1 ) holds for any function quently there is a number 𝜆 such that the relation 𝑥(𝑠 ̃ 𝑥̃ ∈ 𝐸(𝛽𝑆0 ). We show that 𝜆 = 1, i. e., the points 𝑠0 and 𝑠1 are equivalent. This completes the proof of the proposition. Let (𝑠𝛼 ) and (𝑡𝛼 ) be two nets with 𝑠𝛼 ∈ 𝐹𝑛0 , 𝑠𝛼 󳨀→ 𝑠0 and 𝑡𝛼 ∈ 𝐹𝑛1 , 𝑡𝛼 󳨀→ 𝑠1 . For any function 𝑥 ∈ 𝐸(𝑆0 ) one has 𝑥(𝑠𝛼 ) ̃ 0 ) and 󳨀→ 𝑥(𝑠 𝑒(𝑠𝛼)

𝑥(𝑡𝛼 ) ̃ 1 ). 󳨀→ 𝑥(𝑠 𝑒(𝑡𝛼)

In particular, one has lim 𝛼

𝑒𝜈(𝑠𝛼 ) 𝑒𝜈 (𝑡𝛼 ) = 𝜆 lim , 𝛼 𝑒(𝑡 ) 𝑒(𝑠𝛼 ) 𝛼

𝜈 = 1, 2, . . . ,

which yields the equality ∞

∑ 𝜆 𝜈 lim 𝛼

𝜈=1

∞ 𝑒𝜈(𝑠𝛼 ) 𝑒𝜈 (𝑡𝛼) = 𝜆 ∑ 𝜆 𝜈 lim . 𝛼 𝑒(𝑡 ) 𝑒(𝑠𝛼 ) 𝛼 𝜈=1 𝑒 (𝑠 )

𝜈 𝛼 Our choice of the reals 𝜆 𝜈 ensures the uniform convergence of the series ∑∞ 𝜈=1 𝜆 𝜈 𝑒(𝑠𝛼 ) on each of the sets 𝐹𝑛. Therefore the order of summation and taking the limit can be changed. Hence the last equality can be written as

∞

lim (∑ 𝜆𝜈 𝛼 𝜈=1

∞ 𝑒𝜈 (𝑠𝛼 ) 𝑒 (𝑡 ) ( ∑ 𝜆𝜈 𝜈 𝛼 ) . ) = 𝜆 lim 𝛼 𝑒(𝑠𝛼) 𝑒(𝑡𝛼) 𝜈=1

Since by (9.2) the function 𝑒 is 𝑒(𝑠) = ∑∞ 𝜈=1 𝜆 𝜈 𝑒𝜈(𝑠) for 𝑠 ∈ 𝑆0 , we find 𝜆 = 1.

9.2 The existence of Φ𝛼-representations for vector lattices of type (Σ)

| 181

Proposition 9.10. Let 𝐸 be a vector lattice which contains a sequence (𝜑𝑛 )𝑛∈ℕ of finite elements with the property ⋃∞ 𝑛=1 𝐺𝜑𝑛 = M(𝐸). Then the following statements hold: (1) M(𝐸) is locally compact and 𝜎-compact; (2) Φ1 (𝐸) = Φ2 (𝐸); (3) Φ2 (𝐸) is complete in 𝐸; (4) Φ2 (𝐸) is a vector lattice of type (Σ); (5) for each finite element 𝜑 ∈ 𝐸 there is a number 𝑁 such that the element 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨𝜑1 󵄨󵄨 ∨ . . . ∨ 󵄨󵄨𝜑𝑁 󵄨󵄨 is an 𝐸-majorant of 𝜑. Proof. Obviously one has MΦ = M, which implies, according to to Theorem 6.13 (2), the local compactness of M. The statements (2) and (3) of the proposition immediately follow from the Corollaries 6.16 and 6.19. The 𝜎-compactness of the proposition are ∞ proved. The 𝜎-compactness of M follows from the equality ⋃𝑛=1 𝐺𝜑𝑛 = MΦ (𝐸) with respect to the 𝜎-compactness of the sets 𝐺𝜑𝑛 , 𝑛 ∈ ℕ, as proved in Theorem 6.13 (1). Statement (4) is a consequence of Corollary 6.20. The last statement follows from Theorem 6.10 (1) by taking into consideration that supp(𝜑) is a compact subset of M, which enables us to select a finite covering for supp(𝜑) out of the open covering 𝐺𝜑1 , 𝐺𝜑2 , . . . , 𝐺𝜑𝑛 , . . . such that supp(𝜑) ⊂ 𝐺|𝜑1 |∨⋅⋅⋅∨|𝜑𝑁 | . Remark 9.11. If one requires the local compactness and the 𝜎-compactness of M(𝐸) in the previous proposition instead of the existence of a sequence with the properties mentioned, then the existence of such a sequence of finite elements is guaranteed by Corollary 6.20 and the statements (2)–(5) obviously hold. For vector lattices of type (Σ) we now answer the question regarding the existence of Φ𝛼-representations. A slight generalization of the sufficiency part of the next result is provided in Chapter 10, where we get this result in a different framework. Theorem 9.12 (Existence of Φ𝛼-representations). Let 𝐸 be a vector lattice of type (Σ). For the existence of a Φ𝛼-representation (𝐸(𝑄), 𝑖) of 𝐸 on a 𝜎-compact space 𝑄, it is sufficient, and in the case of uniform completeness of 𝐸 also necessary, that a sequence (𝜑𝑛)∞ 𝑛=1 of finite elements in 𝐸 exists with the property ∞

M(𝐸) = ⋃ 𝐺𝜑𝑛 . 𝑛=1

Proof. Necessity. Theorem 8.22 ensures the equality M = MΦ , so due to Remark 6.31, we dispose both of the local compactness and 𝜎-compactness of M. The existence of a sequence of finite elements with the required property is guaranteed now as described in Remark 9.11. Sufficiency. Let (𝑒𝜈)𝜈∈ℕ be a sequence which satisfies the condition (Σ󸀠 ) in 𝐸, and (𝜑𝑛)𝑛∈ℕ be a sequence of finite elements in 𝐸 with ⋃∞ 𝑛=1 𝐺𝜑𝑛 = M(𝐸), where without loss of generality 0 ≤ 𝜑1 ≤ 𝜑2 ≤ ⋅ ⋅ ⋅ ≤ 𝜑𝑛 ≤ ⋅ ⋅ ⋅ may be assumed. Clearly, the system (𝜑𝑛)𝑛∈ℕ is complete in 𝐸, hence by Theorem 9.8 a Riesz norm exists on 𝐸, and by Proposition 9.9 a (⋆)-representation (𝐸(𝑄), 𝑖) exists on some 𝜎-compact space 𝑄, where the

182 | 9 Representations of vector lattices by means of continuous functions images of all elements 𝜑𝑛 are finite functions 𝑖(𝜑𝑛 ), 𝑛 ∈ ℕ. We make sure that (𝐸(𝑄), 𝑖) is the required representation. According to Proposition 9.10, for each finite element 󵄨 󵄨 𝜑0 ∈ 𝐸 one can find a number 𝑛0 and a real 𝐶 > 0 such that 󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨 ≤ 𝐶𝜑𝑛0 . The support of the function 𝑖(𝜑0 ) is therefore contained in the support of the function 𝑖(𝜑𝑛0 ), i. e., {𝑡 ∈ 𝑄 : (𝑖𝜑0 )(𝑡) ≠ 0} ⊂ supp(𝑖𝜑𝑛0 ). which shows that 𝑖𝜑0 is a finite function. This proves the condition (Φ) to hold in the vector lattice 𝐸(𝑄) and, consequently, (𝐸(𝑄), 𝑖) is a Φ-representation of 𝐸. It remains to prove that 𝐸(𝑄) also satisfies the condition (𝛼). For that purpose observe that for any point 𝑡 ∈ 𝑄 there is an index 𝑗𝑡 with 𝜑𝑗𝑡 (𝑡) > 0. Therefore the collection of the finite functions belonging to 𝐸(𝑄) satisfies the condition (⋆), which is the reason for con∞ cluding 𝑄 = ⋃𝑛=1 supp(𝑖𝜑𝑛 ). According to statement (4) of Proposition 9.9, the vector lattice 𝐸(𝑄) separates the points of 𝑄 strongly, therefore, by Proposition 8.3, the vector lattice 𝐸(𝑄) satisfies the condition (𝛼). Remark 9.13. (1) From the proof of the sufficiency is clear that even ∞

𝑄 = ⋃{𝑡 ∈ 𝑄 : 𝜑𝑛 (𝑡) ≠ 0} 𝑛=1

holds; cf. Corollary 8.11. This corollary also shows that 𝑄 and ⋃∞ 𝑛=1 𝐺𝜑𝑛 are homeomorphic. (2) The uniform completeness is essential in the part of necessity; see also the comment on Example 9.5. (3) By taking Remark 9.11 into account, the condition of the theorem can be replaced by the requirement of the local compactness and 𝜎-compactness for the space M(𝐸) (see also Theorem 10.16). (4) The condition of the theorem is exactly the requirement of the existence of a countable sufficient set of finite elements in 𝐸.

9.3 𝐿𝐹-vector lattices In this section, we need the notion of a strict inductive limit 𝑋 of a sequence of subspaces 𝑋𝑛 . We assume that (𝑋1 , 𝜏1 ) ⊂ (𝑋2 , 𝜏2 ) ⊂ ⋅ ⋅ ⋅ ⊂ (𝑋𝑛, 𝜏𝑛 ) ⊂ (𝑋𝑛+1 , 𝜏𝑛+1 ) ⊂ ⋅ ⋅ ⋅ , is a strictly increasing sequence of vector subspaces of 𝑋 with 𝑋 = ⋃𝑛∈ℕ 𝑋𝑛 , where each 𝑋𝑛 is equipped with a locally convex topology 𝜏𝑛 , such that for each 𝑛 the topology 𝜏𝑛+1 induces the topology 𝜏𝑛 on 𝑋𝑛 . The topology on 𝑋 is the strongest locally convex topology 𝜏 on 𝑋 which induces on each 𝑋𝑛 a topology which is majorized by 𝜏𝑛 . The pair (𝑋, 𝜏) is called the strict inductive limit of the sequence (𝑋𝑛, 𝜏𝑛 )𝑛∈ℕ .

9.3 𝐿𝐹-vector lattices

|

183

It is well known that for a strict inductive limit (𝑋, 𝜏), the topology 𝜏 induces the topology 𝜏𝑛 on each subspace 𝑋𝑛 , and that 𝑋 is a Hausdorff complete locally convex vector space if the spaces 𝑋𝑛 are Hausdorff complete locally convex topological vector spaces; see [104, Chap.VII.1]. A strict inductive limit of a sequence of Fréchet-spaces³ is called an 𝐿𝐹-space; see [106, Chap.II.6]. A Fréchet-lattice is a complete metrizable locally convex-solid Riesz space, i. e., a Fréchet-space with a neighborhood basis at zero consisting of solid sets (or equivalently, if the topology is defined by a family of Riesz seminorms). Definition 9.14. A vector lattice is called an 𝐿𝐹-lattice or, more precisely, an 𝐿𝐹-vector lattice, if it is the strict inductive limit of a sequence of Fréchet-lattices. From previous statements it is clear that any 𝐿𝐹-vector lattice is uniformly complete. In his paper [63], I. Kawai proved a result on the representation for 𝐿𝐹- vector lattices of type (Σ). In this section we establish Kawai’s result as a corollary of our Theorem 9.12. First of all, Theorem 9.12 implies the next result. Theorem 9.15. A vector lattice 𝐸 of type (Σ) such that 𝐸 = Φ1 (𝐸) possesses a (Φ𝛼)representation (𝐸(𝑄), 𝑖) on a locally compact, 𝜎-compact space 𝑄. If, in addition, 𝐸 is uniformly complete, then 𝐸(𝑄) coincides with the set K(𝑄) of all finite functions on 𝑄. Proof. For each sequence (𝑒𝜈)𝜈∈ℕ of elements which satisfies the condition (Σ󸀠 ) in 𝐸, one has ⋃∞ 𝜈=1 𝐺𝑒𝜈 = M(𝐸). This shows that the first part of the theorem is an obvious corollary of the previous theorem. The second part immediately follows from Corollary 8.8. Theorem 9.16. All elements of an 𝐿𝐹-vector lattice of type (Σ) are finite. Proof. Let 𝐸 be an 𝐿𝐹-vector lattice of the sequence (𝐸𝑘 )𝑘∈ℕ . If (𝑒𝑛)𝑛∈ℕ is a sequence in 𝐸 which satisfies the condition (Σ󸀠 ), then the subspaces 𝑋𝑛 = {𝑥 ∈ 𝐸 : ∃ 𝜆 > 0, |𝑥| ≤ 𝜆𝑒𝑛}, equipped with the norm ‖𝑥‖𝑛 = inf{𝜆 > 0 : |𝑥| ≤ 𝜆𝑒𝑛},

𝑛 = 1, 2, . . .

are Banach lattices⁴. Since 𝐸 is also the inductive limit of the spaces 𝑋𝑛 , then according to [44, Theorem.6.5.1], for any numbers 𝑛 and 𝑚 there are numbers 𝑝𝑛 and 𝑞𝑚 , such that 𝑋𝑛 ⊂ 𝐸𝑝𝑛

and 𝐸𝑚 ⊂ 𝑋𝑞𝑚 ,

3 A Fréchet space is a complete metrizable locally convex space. 4 Even 𝐴𝑀-spaces with unit 𝑒𝑛 , where [−𝑒𝑛, 𝑒𝑛 ] is the closed unit ball; see p. 14.

184 | 9 Representations of vector lattices by means of continuous functions where the inclusions 𝑋𝑛 󳨅→ 𝐸𝑝𝑛 and 𝐸𝑚 󳨅→ 𝑋𝑞𝑚 are continuous mappings. For the proof of the finiteness of each element of 𝐸 it suffices to establish that the elements 𝑒𝑛, 𝑛 ∈ ℕ are finite. This will be done only for 𝑒1 , since for all other elements 𝑒𝑛 with 𝑛 ≥ 2 the proof is analogous. Therefore we shall prove the existence of some number 𝑛1 such that the element 𝑒𝑛1 is an 𝐸-majorant for 𝑒1 . For an arbitrary fixed element 𝑥 ∈ 𝐸, we will show that the set 𝐴 𝑥 = {|𝑥| ∧ 𝑐 𝑒1 : 𝑐 ≥ 0} is bounded in the space 𝑋𝑛1 , i. e., that 𝐴 𝑥 ⊂ 𝜆 𝑥 [−𝑒𝑛1 , 𝑒𝑛1 ] for some 𝜆 𝑥 > 0. Of course, this implies the finiteness of 𝑒1 . Clearly, 𝐴 𝑥 ⊂ 𝑋1 ⊂ 𝐸𝑝1 and moreover, 𝐴 𝑥 is a bounded subset of 𝑋𝑛 if 𝑥 belongs to 𝑋𝑛 . Then 𝐴 𝑥 is also bounded in 𝐸𝑝𝑛 . Since the topology in 𝐸𝑝1 is nothing more than the restriction to 𝐸𝑝1 of the topology in 𝐸𝑝𝑛 , the subset 𝐴 𝑥 is also bounded in 𝐸𝑝1 . From this, the boundedness of 𝐴 𝑥 in the space 𝑋𝑞𝑝 follows immediately, and so 𝑛1 = 𝑞𝑝1 is 1 the required number. The last two theorems yield the following. Theorem 9.17 (I. Kawai [63, Theorem 6.6]). Any 𝐿𝐹-vector lattice 𝐸 of type (Σ) possesses a Φ𝛼-representation (𝐸(𝑄), 𝑖) on some locally compact 𝜎-compact space 𝑄, where, due to uniform completeness, 𝐸(𝑄) coincides with the set K(𝑄) of all finite functions on 𝑄.

9.4 Vector lattices of type (𝐶𝑀 ) In Section 8.2 we introduced the notion of a vector lattice of slowly growing functions. In the present section we define and study a class of vector lattices which might be considered the abstract analogue of that class of function vector lattices. In the sequel we prove the existence of Φ𝛼-representations as vector lattices of slowly growing functions for this new class of vector lattices. Definition 9.18. Let 𝐸 be a vector lattice and Φ1 (𝐸) the ideal of all finite elements of 𝐸. We say that 𝐸 satisfies the condition (𝜇) if for each element 𝑥 ∈ 𝐸, an element 𝑦 ∈ 𝐸 exists such that for any 𝜀 > 0 there is an element 0 < 𝑥𝜀 ∈ Φ1 (𝐸) with the property |𝑥| ≤ 𝑥𝜀 ∨ 𝜀 𝑦.

(9.3)

A vector lattice of type (Σ) which satisfies the condition (𝜇) is called a vector lattice of type (𝐶𝑀 ). The condition (𝜇) holds in each vector lattice of slowly growing functions 𝐶𝑀 (𝑄). In order to show the implication (𝐶𝑀 ) 󳨐⇒ (𝜇), (9.4) it suffices to establish the Inequality (9.3) only for the elements of a sequence (𝑒𝜈)𝜈∈ℕ which satisfies in 𝐶𝑀 (𝑄) the condition (Σ󸀠). Indeed, due to condition (𝐶𝑀 ), for each 𝜈 ∈ ℕ and 𝜀 > 0 there is some 𝑘𝜈 that 𝑒𝜈 (𝑡) ≤ 𝜀 e𝑘𝜈 (𝑡) holds outside of some compact

9.4 Vector lattices of type (𝐶𝑀 )

| 185

subset 𝐾 ⊂ 𝑄 which depends on 𝜈 and 𝜀. By means of condition (𝛼), a finite function 𝜑𝜀 can be constructed such that 𝜑𝜀 (𝜈) ≥ max𝑠∈𝐾 𝑒𝜈 (𝑠) for 𝑡 ∈ 𝐾. Thus the implication is proved. We do not claim that a vector lattice of majorizing functions which satisfies the condition (𝜇) is also a vector lattice of type (𝐶𝑀 ), as Example 8.15 shows. As already mentioned (see p. 164), the vector lattice 𝐸 = 𝐸([1, ∞) × [1, ∞)) in that example is not a vector lattice of slowly growing functions, since the property (𝐶𝑀 ) is not satisfied. However, the condition (𝜇) holds. As mentioned before, the latter fact has to be shown only for the elements 𝑒𝜈 (𝑡, 𝑠) = 𝑡𝜈 of the sequence which satisfies the condition (Σ󸀠 ) in 𝐸. We give a short proof only for 𝑒1 . For given 𝜀 outside the compact interval [1, 1𝜀 ], one has 𝑒1 (𝑡, 𝑠) = 𝑡 ≤ 𝜀 𝑡2 = 𝜀 𝑒2 (𝑡, 𝑠). Define {𝑒1 (𝑡, 𝑠), { { 𝜑𝜀 (𝑡) = {linear, { { {0,

𝑡 ∈ [1, 1𝜀 ] 𝑡 ∈ [ 1𝜀 , 2𝜀 ] 𝑡 ∈ [ 2𝜀 , ∞)

as a continuous function and take 𝑦 = 𝑒2 . Then 𝜑𝜀 ∈ Φ1 (𝐸) and 𝑒1 ≤ 𝜑𝜀 ∨ 𝜀𝑒2 . Nevertheless, the following result holds; see [89]. Theorem 9.19. Let 𝐸 be a vector lattice of type (𝐶𝑀 ). Then the following statements hold: (1) a Φ𝛼-representation of 𝐸 exists on a 𝜎-compact space 𝑄; (2) each Φ𝛼-representation of 𝐸 is a vector lattice of slowly growing functions. If 𝐸 is uniformly complete, then 𝐸(𝑄) is a solid subset of 𝐶(𝑄). Proof. If (𝑒𝜈 )𝜈∈ℕ is a sequence which satisfies the condition (Σ󸀠) in 𝐸, then for each number 𝜈 there is a number 𝑚𝜈 , such that for any natural 𝑘 a finite element 𝜑𝜈,𝑘 ∈ 𝐸 exists with 𝑒𝜈 ≤ 𝜑𝜈,𝑘 ∨ 1𝑘 𝑒𝑚𝜈 . We show M(𝐸) = ⋃∞ 𝜈,𝑘=1 𝐺𝜑𝜈,𝑘 . If a discrete functional 𝑓 vanishes on all elements 𝜑𝜈,𝑘 for 𝜈, 𝑘 ∈ ℕ, then 𝑓(𝑒𝜈 ) ≤ 1𝑘 𝑓(𝑒𝑚𝜈 ) for all 𝑘, and so 𝑓(𝑒𝜈 ) = 0. Since 𝜈 was arbitrary, we get 𝑓 = 0, which contradicts the definition of a discrete functional⁵. According to Theorem 9.12, a Φ𝛼-representation (𝐸(𝑄), 𝑖) on a 𝜎compact (and locally compact) space 𝑄 exists which is homeomorphic to M. In order to show that 𝐸(𝑄) is a vector lattice of slowly growing functions, notice that the relation 𝑒𝜈 ≤ 𝜑𝜀 ∨ 𝜀 𝑒𝑚𝜈 in 𝐸 (i. e., condition (𝜇) applied to the element 𝑒𝜈 ) transforms (under 𝑖) into the relation for continuous functions⁶ 𝑒𝜈(𝑡) ≤ 𝜑𝜀 (𝑡) ∨ 𝜀 𝑒𝑚𝜈 (𝑡) for all 𝑡 ∈ 𝑄. This shows that outside the support of the function 𝜑𝜀 , which, due to the condition (Φ), is a compact set, the inequality 𝑒𝜈(𝑡) ≤ 𝜀 𝑒𝑚𝜈 (𝑡) holds. Our argument ensures that any Φ𝛼-representation of 𝐸 is a vector lattice of slowly growing functions as well.

5 Thus it is proved that for any maximal ideal 𝑀 ∈ M(𝐸), there is at least one 𝜑𝜈,𝑘 with 𝑀 ∈ 𝐺𝜑𝜈,𝑘 . 6 For simplicity, the image in 𝐸(𝑄) of the element 𝑥 ∈ 𝐸 under the isomorphism 𝑖 is also denoted by 𝑥.

186 | 9 Representations of vector lattices by means of continuous functions For the proof of the last assertion of the second statement of the theorem, assume 0 ≤ 𝑔 ≤ 𝑥 for arbitrary functions 𝑔 ∈ 𝐶(𝑄) and 𝑥 ∈ 𝐸(𝑄). Without loss of generality we suppose 𝑥 ≤ 𝑒1 and take 𝑒2 instead of 𝑒𝑚1 , where 𝑚1 is the number which is found for 𝑒1 in the first part of the proof. Due to the uniform completeness of 𝐸, and in view of Corollary 8.8, the collection 𝐸(𝑄) contains all continuous finite functions on 𝑄, in particular the continuous functions 𝑦𝑗 with 0 ≤ 𝑦1 ≤ 𝑦2 ≤ . . . and { {max 𝑒𝑗(𝑠), 𝑡 ∈ 𝑄𝑗 𝑦𝑗 (𝑡) = { 𝑠∈𝑄𝑗 , 𝑗 ∈ ℕ, { 0, 𝑡 ∈ 𝑄 \ 𝑄𝑗+1 { where 𝑄𝑗 ⊂ 𝑄 are compact subsets such that 𝑄𝑗 ⊂ int(𝑄𝑗+1 ) and 𝑄 = ⋃∞ 𝑗=1 𝑄𝑗 . The functions 𝑎𝑗 = 𝑔 ∧ 𝑦𝑗 , 𝑗 ∈ ℕ are finite, belong to 𝐸(𝑄) and satisfy the inequalities 0 ≤ 𝑎1 ≤ 𝑎2 ≤ . . .. For each compact subset 𝐾 ⊂ 𝑄 there is an index 𝑗0 with 𝑎𝑗 (𝑡) = 𝑔(𝑡),

𝑡∈𝐾

for all 𝑗 ≥ 𝑗0 .

For arbitrary fixed 𝜀 > 0, select 𝜑0 ∈ Φ1 (𝐸) such that 𝑒1 ≤ 𝜑0 ∨ 𝜀𝑒2 . Then for appropriate 𝑗1 one has supp(𝑖𝜑0 ) ⊂ 𝑄𝑗1 −1

and

𝑎𝑗1 +𝑝 − 𝑎𝑗1 ≤ 𝜀 𝑒2 , for all 𝑝 ∈ ℕ.

The uniform completeness of 𝐸 guarantees the existence of a function 𝑎 in 𝐸(𝑄) with 𝑎 = (𝑟)-lim 𝑎𝑗 . Since on the compact subsets of 𝑄 the functions 𝑎 and 𝑔 coincide, 𝑔 ∈ 𝐸(𝑄) is established. Remark 9.20. (1) As we see from the first part of the proof, in any vector lattice of type (𝐶𝑀 ) a countable sufficient number of finite elements exists (see Remark 9.13 (3)). We have also shown that M(𝐸) = ⋃∞ 𝜈,𝑘=1 𝐺𝜑𝜈,𝑘 , where 𝜑𝜈,𝑘 ∈ Φ1 (𝐸) for all 𝜈, 𝑘 ∈ ℕ. So M(𝐸) = MΦ (𝐸) and one can say that Φ𝛼-representations of vector lattices of type (𝐶𝑀 ) arise on spaces homeomorphic to MΦ . (2) In Section 10.1 we will see that in the case of a vector lattice of type (𝐶𝑀 ), the set Φ1 (𝐸) is a so-called 𝜎-base of finite elements in 𝐸. If, in addition, 𝐸 is uniformly complete, then Φ1 (𝐸) is even a normal 𝜎-base. The existence of a Φ𝛼representation just proved in the theorem can also be treated as a representation by means of the base Φ1 (𝐸); see Section 10.2. Without uniform completeness the vector sublattice 𝐸(𝑄) may not be solid in 𝐶(𝑄), as the next example demonstrates. Example 9.21. A vector lattice 𝐸(𝑇) of type (Σ) consisting of continuous functions which satisfy the condition (𝜇), but 𝐸(𝑇) is not solid in 𝐶(𝑇); see also Example 10.21. Consider on 𝑇 = [1, +∞) the set of all continuous piecewise polynomials, i. e., for each function belonging to 𝐸 = 𝐸(𝑇), the set 𝑇 is split into a finite number of intervals, on each of which the function is a polynomial; cf. Example 3.64. This vector lattice is of type (Σ), where for a sequence which satisfies the condition (Σ󸀠 ) in 𝐸, one can take the restrictions

9.4 Vector lattices of type (𝐶𝑀 ) |

187

of the polynomials (𝑡𝜈 )𝜈∈ℕ on 𝑇. It is clear that 𝐸 has the following properties: (a) K(𝑇) ∩ 𝐸(𝑇) ⊂ Φ1 (𝐸); (b) 𝐸 satisfies the condition (𝜇); (c) 𝐸 is not a solid subspace in 𝐶(𝑇), and moreover, several finite functions do not belong to 𝐸(𝑇); (d) 𝐸(𝑇) is not uniformly complete. Definition 9.22. A subset 𝐴 of a vector lattice 𝐸 is said to be (𝑟)-dense in 𝐸 if each element 𝑥 ∈ 𝐸 is the uniform limit of a sequence of elements of 𝐴, i. e., for each element 𝑥 a sequence (𝑎𝑛 )𝑛∈ℕ , 𝑎𝑛 ∈ 𝐴 and a regulator 𝑢 ∈ 𝐸 exist, such that for any 𝜀 > 0 a 󵄨 󵄨 number 𝑛0 (𝜀) exists such that the inequality 󵄨󵄨󵄨𝑎𝑛 − 𝑥󵄨󵄨󵄨 ≤ 𝜀 𝑢 holds for all 𝑛 ≥ 𝑛0 (𝜀). In the next theorem it is established that the (𝑟)-density of Φ1 (𝐸) is equivalent to the condition (𝜇). Theorem 9.23. In a vector lattice 𝐸 the condition (𝜇) holds if and only if each element of 𝐸 is the uniform limit of sequence of finite elements. Proof. Necessity. Let the condition (𝜇) be satisfied in the vector lattice 𝐸. We show that any element of 𝐸 is the uniform limes of a sequence of finite elements, where it is clear that we may restrict ourselves to positive elements. So for 0 ≤ 𝑥 ∈ 𝐸 there is an element 𝑦 ∈ 𝐸, and for arbitrary 𝜀 > 0 an element 𝜑𝜀 ∈ Φ1 (𝐸) such that 𝑥 ≤ 𝜑𝜀 ∨ 𝜀 𝑦 ≤ 𝜑𝜀 + 𝜀 𝑦 1 ≤ 𝜀. Then for each 𝑘 ≥ 𝑘0 𝑘0 1 one can find an element 𝜑𝑘 ∈ Φ1 (𝐸) with 𝑥 ≤ 𝜑𝑘 + 𝑘 𝑦. Due to the Riesz decomposition property (see p. 5 and Section 2.1), the element 𝑥 has a representation 𝑥 = 𝑥󸀠𝑘 + 𝑥󸀠󸀠𝑘 , where 0 ≤ 𝑥󸀠𝑘 ≤ 𝜑𝑘 and 0 ≤ 𝑥󸀠󸀠𝑘 ≤ 𝑘1 𝑦 ≤ 𝜀 𝑦. It is clear that 𝑥󸀠𝑘 are finite elements for

holds. Fix now an 𝜀 > 0 and a natural number 𝑘0 with

𝑘 ≥ 𝑘0 . The relations

󵄨 󵄨󵄨 󵄨󵄨𝑥 − 𝑥󸀠𝑘 󵄨󵄨󵄨 = 𝑥󸀠󸀠𝑘 ≤ 𝜀 𝑦, 󵄨 󵄨

which hold for sufficiently large 𝑘, mean that the element 𝑥 is the uniform limes of the sequence (𝑥󸀠𝑘 )𝑘∈ℕ≥𝑘 . 0 Sufficiency. We fix an arbitrary positive element 𝑥 ∈ 𝐸, which by assumption is the uniform limit of a sequence of finite elements with some regulator 𝑦. Therefore, for an arbitrary 𝜀 > 0, there is a finite element 𝑥0 with 󵄨󵄨 󵄨 󵄨󵄨𝑥 − 𝑥0 󵄨󵄨󵄨 ≤ 𝜀 𝑦. The condition (𝜇) follows immediately from the estimates 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑥 ≤ 󵄨󵄨󵄨𝑥 − 𝑥0 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑥0 󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝑥0 󵄨󵄨󵄨 + 𝜀 𝑦 ≤ 2(󵄨󵄨󵄨𝑥0 󵄨󵄨󵄨 ∨ 𝜀 𝑦) = 2 󵄨󵄨󵄨𝑥0 󵄨󵄨󵄨 ∨ 2𝜀 𝑦. In the remaining part of this section we give a new approach to vector lattices of type (𝐶𝑀), see [124].

188 | 9 Representations of vector lattices by means of continuous functions Theorem 9.24. A vector lattice 𝐸 of type (Σ) is of type (𝐶𝑀 ) if and only if some (and consequently any) sequence (𝑒𝜈)𝜈∈ℕ of elements which satisfies the condition (Σ󸀠 ) in 𝐸 possesses the following properties: In 𝐸 three sequences of positive elements (𝜎𝜈󸀠 )𝜈∈ℕ ,

(𝜎𝜈 )𝜈∈ℕ ,

(𝜎𝜈 )𝜈∈ℕ

exist such that 𝑒𝜈 = 𝜎𝜈 + 𝜎𝜈󸀠 + 𝜎𝜈 { { { { { {𝜎𝜈 ∈ Φ1 (𝐸) 𝜈 ∈ ℕ; (a) { { 𝜎𝜈 ⊥ 𝜎𝜈 { { { { 󸀠 {𝜎𝜈 + 𝜎𝜈 ≤ 𝜎𝜈+1 (b) for any 𝜀 > 0 and each 𝜈 ∈ ℕ there is an index 𝑘 = 𝑘(𝜀, 𝜈) with 𝜎𝜈 ∨ 𝜎𝑘 − 𝜎𝑘 ≤ 𝜀 𝑒𝜈+1 . Proof. Let 𝐸 be a vector lattice of type (𝐶𝑀 ) and (𝑒𝜈)𝜈∈ℕ a fixed sequence which satisfies the condition (Σ󸀠 ) in 𝐸. Let (𝐸(𝑄), 𝑖) be a Φ𝛼-representation⁷ whose existence is guaranteed by Theorem 9.19. The condition (𝜇) holds in 𝐸 due to the type (𝐶𝑀 ), where without loss of generality we assume (𝜇) to be satisfied in the following form: for any 𝜈 and each 𝜀 > 0 there is a finite element 𝜑0 such that 𝑒𝜈 ≤ 𝜑0 ∨ 𝜀 𝑒𝜈+1 . Let 𝜑𝑘(𝜈) be positive finite elements which satisfy the following inequalities 𝑒𝜈 ≤ 𝜑𝑘(𝜈) ∨

1 𝑘

𝑒𝜈+1

for

𝑘, 𝜈 ∈ ℕ.

In 𝐸(𝑄), apart from the finite function 𝜑2(1), a finite function 𝑥1 is found such that 𝑒1 ≤ 𝜑2(1) ∨

1 2

𝑒2

and 𝑥1 (𝑡) ≥ 1

for

𝑡 ∈ supp(𝜑2(1)).

Denote 𝐾1 = supp(𝜑2(1)). Let 𝑙1 be the smallest natural number 𝑙 which satisfies the condition 𝑙 ≥ max𝑡∈𝐾1 𝑒1 (𝑡). On the compact set 𝐾1 there holds 𝑒1 (𝑡) ∧ 𝑙1 𝑥1 (𝑡) = 𝑒1 (𝑡) and outside 𝐾1 one has 𝜑2(1) (𝑡) = 0, i. e., (𝑒1 − 𝑒1 ∧ 𝑙1 𝑥1 ) ⊥ 𝜑2(1) . Put 𝜎1 := 𝜑2(1) ∧ 𝑒1 ,

𝜎1 := 𝑒1 − 𝑒1 ∧ 𝑙1 𝑥1 ,

𝜎1󸀠 := 𝑒1 − (𝜎1 + 𝜎1 ).

Then 𝜎1 ⊥ 𝜎1 . For the function 𝜑3(2) , the inequality 𝑒2 ≤ 𝜑3(2) ∨ 13 𝑒3 holds. Denote by 𝜑2 the finite element 𝑙1 𝑥1 ∨ 𝜑3(2) ∨ 𝜑3(1) , and by 𝐾2 the set supp(𝜑2 ). Then 𝑒1 ≤ 𝜑2 ∨

1 3

𝑒2

and

𝑒2 ≤ 𝜑2 ∨

1 3

𝑒3 .

Let 𝑥2 be a finite function of 𝐸(𝑄) with 𝑥2 (𝑡) ≥ 1 for 𝑡 ∈ 𝐾2 , and let 𝑙2 ∈ ℕ be the smallest 𝑙 with 𝑙 ≥ max𝑡∈𝐾2 𝑒2 (𝑡). Put 𝜎2 := 𝜑2 ∧ 𝑒2 ,

𝜎2 := 𝑒2 − 𝑒2 ∧ 𝑙2 𝑥2 ,

𝜎2󸀠 := 𝑒2 − (𝜎2 + 𝜎2 ).

Then 𝜎1 + 𝜎1󸀠 = 𝜎1 + 𝑒1 − 𝜎1 − 𝜎1 = 𝑒1 ∧ 𝑙1 𝑥1 ≤ 𝑒2 ∧ 𝜑2 ,

7 For simplicity, we will denote the image in 𝐸(𝑄) under 𝑖 of the element 𝑥 ∈ 𝐸 again by 𝑥.

9.4 Vector lattices of type (𝐶𝑀 )

|

189

and so 𝜎1 + 𝜎1󸀠 ≤ 𝜎2 and 𝜎2 ⊥ 𝜎2 . The last holds, due to (𝑒2 − 𝑒2 ∧ 𝑙2 𝑥2 ) ⊥ 𝜑2 . For 𝜈 > 2 the elements 𝜎𝜈 , 𝜎𝜈󸀠 , 𝜎𝜈 are constructed analogously as follows: first find a finite element 1 𝜑𝜈 such that 𝑒𝜈 ≤ 𝜑𝜈 ∨ 𝜈+1 𝑒𝜈+1 , where it may be assumed that the inequalities 𝑒𝑘 ≤ 𝜑𝜈 ∨

1 𝑒 𝜈+1 𝑘+1

for 𝑘 = 1, 2, . . . , 𝜈 − 1

and 𝜑𝜈 ≥ 𝑙𝜈−1 𝑥𝜈−1

already hold. Next construct a finite function 𝑥𝜈 ∈ 𝐸(𝑄) which takes on values greater than 1 on the compact set supp(𝜑𝜈 ). The number 𝑙𝜈 is then defined such that (𝑒𝜈 − 𝑒𝜈 ∧ 𝑙𝜈 𝑥𝜈 ) and 𝜑𝜈 are disjoint functions. Finally put 𝜎𝜈 := 𝜑𝜈 ∧ 𝑒𝜈 ,

𝜎𝜈 := 𝑒𝜈 − 𝑒𝜈 ∧ 𝑙𝜈 𝑥𝜈 ,

𝜎𝜈󸀠 := 𝑒𝜈 − (𝜎𝜈 + 𝜎𝜈 ).

From the construction described, it is clear that for each 𝜈 the element 𝑒𝜈 has the decomposition 𝑒𝜈 = 𝜎𝜈 + 𝜎𝜈󸀠 + 𝜎𝜈 with the properties 𝜎𝜈 ∈ Φ1 (𝐸),

𝜎𝜈 , 𝜎𝜈󸀠 , 𝜎𝜈 ≥ 0,

𝜎𝜈 ⊥ 𝜎𝜈 ,

𝜎𝜈 + 𝜎𝜈󸀠 ≤ 𝜎𝜈+1 .

Hence condition (a) is proved. If arbitrary numbers 𝜀 > 0 and 𝜈 ∈ ℕ are fixed, then there is a number 𝑘 ≥ 𝜈+1 such that 𝑘1 ≤ 𝜀. In view of condition (𝜇) and the construction above, the inequalities 𝑒𝜈 ≤ 𝜎𝑘 ∨ 1𝑘 𝑒𝜈+1 ≤ 𝜎𝑘 + 1𝑘 𝑒𝜈+1 hold, and therefore 𝑒𝜈 − 𝜎𝑘 ≤ 𝜀 𝑒𝜈+1 , which implies (𝑒𝜈 − 𝜎𝑘 )+ ≤ 𝜀 𝑒𝜈+1 . Since (𝜎𝜈 − 𝜎𝑘 )+ ≤ (𝑒𝜈 − 𝜎𝑘 )+ , we obtain 𝜎𝜈 ∨ 𝜎𝑘 − 𝜎𝑘 = (𝜎𝜈 − 𝜎𝑘 )+ ≤ 𝜀 𝑒𝜈+1 . Thus it is proved that the sequence (𝜎𝜈 )𝜈∈ℕ of finite elements satisfies condition (b). The inverse also holds. If, in a vector lattice of type (Σ), conditions (a) and (b) are satisfied, then the condition (𝜇) holds. Indeed, for arbitrary 𝜀 > 0 and 𝜈 take 𝑘 ≥ 𝜈 + 1 such that 𝜎𝜈 ∨ 𝜎𝑘 − 𝜎𝑘 ≤ 2𝜀 𝑒𝜈+1 . Then one has 𝑒𝜈 = 𝜎𝜈 + 𝜎𝜈󸀠 + 𝜎𝜈 = 𝜎𝜈󸀠 + 𝜎𝜈 ∨ 𝜎𝜈 ≤ 𝜎𝜈󸀠 + 𝜎𝑘 + (𝜎𝜈 ∨ 𝜎𝑘 − 𝜎𝑘 ) ≤ 2𝜎𝑘 + (𝜎𝜈 ∨ 𝜎𝑘 − 𝜎𝑘 ) ≤ 4𝜎𝑘 ∨ 2(𝜎𝜈 ∨ 𝜎𝑘 − 𝜎𝑘 ) ≤ 4𝜎𝑘 ∨ 𝜀 𝑒𝜈+1 , i. e., 𝑒𝜈 ≤ 4𝜎𝑘 ∨ 𝜀 𝑒𝜈+1 .

(9.5)

Since the element 𝜎𝑘 is finite, the condition (𝜇) follows from the last estimate. Observe the following: if the sequence (𝜎𝜈 )𝜈∈ℕ , which satisfies in 𝐸 conditions (a) and (b) is fixed, then the set {𝑥 ∈ 𝐸 : ∃ 𝑐 > 0 and ∃ 𝜈 with |𝑥| ≤ 𝑐𝜎𝜈 } is the collection Φ1 (𝐸) of all finite elements in 𝐸. This immediately follows from Proposition 9.10 (5), since, due to (9.5), the set (𝜎𝜈 )𝜈∈ℕ is sufficient in 𝐸; see also the proof of Theorem 9.19.

190 | 9 Representations of vector lattices by means of continuous functions Example 9.25. A vector lattice of continuous functions on a topological space 𝑇 is its own e-representation. However, it is not an E-representation⁸. Consider on 𝑇 = [1, ∞) the vector sublattice 𝐸(𝑇) of 𝐶𝑀 (𝑇, (𝑡𝑛)𝑛∈ℕ ) (see Section 8.3), consisting of all functions 𝑥 which satisfy the condition 𝑥(1) = 12 𝑥(2). Then 𝐸(𝑇) satisfies the conditions (⋆) and (𝜇) such that according to Remark 9.20 (1), each discrete functional 𝑓 does not vanish at the ideal of the finite elements. Due to Theorem 8.9, the functional 𝑓 is proportional to a functional 𝛿𝑡 for some 𝑡 ∈ 𝑇. So 𝐸(𝑇) is its own e-representation, however fails to be an E-representation since the standard map 𝜅 is not bijective: the points 𝑡 = 1 and 𝑡 = 2 generate identical maximal ideals. Now are able to provide the example which was announced in Remark 7.23. Example 9.26. A vector lattice with a Riesz norm, such that no discrete functional is continuous, and consequently no maximal ideal is closed with respect to this norm. On 𝑇 = [1, +∞) consider the space of slowly growing functions 𝐸 = 𝐶𝑀 (𝑇, (𝑡𝑛)𝑛∈ℕ ). The vector lattice 𝐸 possesses the following properties (a) condition (𝜇) is satisfied; (b) condition (Φ) is satisfied; see Proposition 8.18; (c) according to Theorem 8.22, the spaces 𝑇 and M(𝐸) are homeomorphic. This implies that each discrete functional 𝑓 ∈ Δ(𝐸) is proportional to a functional of the kind 𝛿𝑡 for some point 𝑡 ∈ 𝑇. If on 𝐸 the monotone norm ∞ −𝑡 ‖𝑥‖ = ∫ |𝑥| e d𝑡 1

is considered, then the functional 𝛿𝑡 is not continuous with respect to that norm for any 𝑡 ∈ 𝑇. Indeed, fix an arbitrary point⁹ 𝑡0 > 1. Then the continuous functions 0, { { { { { {𝑛(𝑡 − 𝑡0 ) + 1, 𝑥𝑛(𝑡) = { {𝑛(𝑡 − 𝑡) + 1, { 0 { { { 0, {

[1, 𝑡0 − 𝑛1 ) [𝑡0 − 𝑛1 , 𝑡0 ) [𝑡0 , 𝑡0 + 1𝑛 ) [𝑡0 + 𝑛1 , ∞)

belong to 𝐸, and 𝑥𝑛 󳨀→ 0 with respect to the norm. However, in view of 𝛿𝑡0 (𝑥𝑛) = 1 for 𝑛→∞ all 𝑛 ∈ ℕ, the functional 𝛿𝑡0 is not continuous.

8 This example was announced in 7.1 after Definition 7.8 9 For 𝑡0 = 1 similar (“one-sided”) functions 𝑥𝑛 are constructed.

10 Representations of vector lattices by means of bases of finite elements In this chapter we deal with representations of vector lattices on locally compact spaces 𝑄, where a special given system of finite elements (later called basis) will be represented as finite functions on 𝑄. The representations considered in this chapter generalize the Φ𝛼-representations which represent all finite elements as finite functions. Besides the existence of basis representations (Theorem 10.10), we get the following important result, that a uniformly complete vector lattice which consists only of finite elements is uniquely defined by its space of maximal ideals if it is isomorphically embeddable into a vector lattice of bounded elements (Corollary 10.12). Example 10.7 shows that without the last condition the results fails. By means of bases of finite elements it is possible to investigate some questions concerning further topological properties of the space 𝑄, if 𝐸(𝑄) is a representation of a vector lattice 𝐸. The normality of embedding 𝐸(𝑄) into 𝐶(𝑄) is answered by means of normal bases (Theorem 10.19). This chapter, of course, is closely related to the two previous Chapters 8 and 9. For example, Theorem 9.12 can again be obtained using the methods developed in this chapter. Again, all vector lattices are assumed to be Archimedean and radical-free. The underlying topological spaces in the representations now turn out to be homeomorphic to subspaces of M(𝐸).

10.1 Bases of finite elements and 𝛼-representations After the next theorem we are able to classify certain ideals of totally finite elements in vector lattices. These ideals (bases) turn out to be useful in the construction of some more qualified representations of vector lattices rather than those studied in Chapter 9; see [92]. Theorem 10.1. Let (𝐸(𝑄), 𝑖) be an 𝛼-representation of the vector lattice 𝐸 and 𝐸0 = {𝑥 ∈ 𝐸 : 𝑖𝑥 ∈ K(𝑄)}. Then (1) 𝐸0 is an ideal of 𝐸 and 𝐸0 ⊂ Φ2 (𝐸); (2) 𝐸0 is complete¹ in 𝐸, i. e., 𝑥 ⊥ 𝑦 for all 𝑥 ∈ 𝐸0 implies 𝑦 = 0; (3) 𝐸0 is embeddable into a vector lattice with order unit (of bounded elements); (4) 𝑄 is homeomorphic to 𝐺(𝐸0 ) (and also to M(𝐸0 )).

1 Or equivalent (for ideals): The ideal 𝐸0 is order dense in 𝐸, i. e., for each 0 < 𝑥 ∈ 𝐸 there is an element 𝑥0 ∈ 𝐸0 such that 0 < 𝑥0 ≤ 𝑥; see [144, Theorem 23.3].

192 | 10 Representations of vector lattices by means of bases of finite elements Proof. The set 𝐸0 = {𝑥 ∈ 𝐸 : 𝑖𝑥 ∈ K(𝑄)} is obviously an ideal in 𝐸. It is favorable to show the final statement of the theorem first. Since 𝐸(𝑄) is a vector lattice that satisfies the condition (𝛼), the subset 𝐸0 (𝑄) of all finite functions of 𝐸(𝑄) satisfies the conditions of Theorem 8.13. So we may conclude that the topological space 𝑄 is homeomorphic to M(𝐸0 ) and therefore, by means of Theorem 6.21 (4), 𝑄 and 𝐺(𝐸0 ) are homeomorphic. In order to prove the remaining part of statement (1), we have to show that any element of 𝐸0 is totally finite. Due to condition (𝛼), the function 𝑖𝑥 for arbitrary 𝑥 ∈ 𝐸0 is a finite element in 𝐸(𝑄) and therefore 𝑥 is also a finite element in 𝐸. According to the already proved statement (4), the sets {𝑡 ∈ 𝑄 : 𝑥(𝑡) ≠ 0} and

{𝑀 ∈ M(𝐸) : 𝑥 ∉ 𝑀}

can be identified, i. e., we may assume supp(𝑥) = suppM (𝑥), where the last set is obviously compact. According to Proposition 8.4, an element 𝑥0 ∈ 𝐸0 with suppM (𝑥) ⊂ 𝐺𝑥0 exists. Since the element 𝑥0 is finite, the inclusion is equivalent to the total finiteness of 𝑥, thus completing the proof of 𝐸0 ⊂ Φ2 (𝐸). (2) In order to establish the completeness of 𝐸0 in 𝐸, it is sufficient (by Corollary 5.15) to show that only the zero-element of 𝐸 belongs to all maximal ideals of the set 𝐺(𝐸0 ). If 𝑥 belonged to all 𝑀 ∈ 𝐺(𝐸0 ), then the function 𝑖𝑥 would vanish at each point of 𝑄, i. e., 𝑥 = 0. (3) Is obvious, since 𝐸0 is isomorphically embeddable into the vector lattice of all bounded continuous functions on 𝑄. In order to provide an answer to several problems concerning special representations of a vector lattice 𝐸, the existence of qualified ideals of totally finite elements in 𝐸, which might be embedded into a vector lattice of bounded elements, is of significant importance; see Theorem 7.21. Based on the previous theorem we introduce some new notations. Definition 10.2. Let 𝐸 be a vector lattice. – an ideal 𝐸0 ⊂ 𝐸 is called a Φ-basis if it consists of totally finite elements of 𝐸 and is complete in 𝐸; – an ideal 𝐸0 ⊂ 𝐸 is called an 𝑅-basis (representation basis) if 𝐸0 satisfies the first three conditions of Theorem 10.1; – an ideal 𝐸0 ⊂ 𝐸 is called a 𝜎-basis if it is a Φ-basis and (considered a vector lattice) is of type (Σ). An ideal 𝐸0 ⊂ 𝐸 is a Φ-basis if it satisfies the first two conditions of Theorem 10.1, and is an 𝑅-basis if it is a Φ-basis and is embeddable into a vector lattice of bounded elements.

10.1 Bases of finite elements and 𝛼-representations

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Remark 10.3. Let 𝐸0 be an 𝑅-basis of the vector lattice 𝐸 and 𝑌 a vector lattice of bounded elements into which 𝐸0 can be embedded. Let 𝐺𝐸 (𝐸0 ) = ⋃ {𝑀 ∈ M(𝐸) : 𝑥 ∉ 𝑀} and 𝑥∈𝐸0

𝐺𝑌 (𝐸0 ) = ⋃ {𝑀 ∈ M(𝑌) : 𝑥 ∉ 𝑀}. 𝑥∈𝐸0

Then, according to Theorem 6.21 (4), the topological spaces 𝐺𝐸 (𝐸0 ), 𝐺𝑌 (𝐸0 ), and M(𝐸0 ) are homeomorphic. It turns out that some internal properties of the ideal 𝐸0 in the vector lattice 𝐸 ensure the embeddability of 𝐸0 into a vector lattice (with an order unit) of bounded elements. Proposition 10.4. Let 𝐸0 be a vector lattice of type (Σ) which consists of finite elements. Then 𝐸0 is embeddable into a vector lattice of bounded elements. Proof. Due to 𝐸0 = Φ1 (𝐸0 ), and since 𝐸0 is of type (Σ), the vector lattice possesses a countable complete collection of finite elements² such that, in view of Theorems 9.8 and 7.21, the vector lattice 𝐸0 is embeddable into a vector lattice of bounded elements. Corollary 10.5. (1) Let 𝐸0 be an ideal of a vector lattice 𝐸 such that 𝐸0 ⊂ Φ2 (𝐸) and 𝐸0 itself is of type (Σ). Then 𝐸0 is embeddable into a vector lattice of bounded elements. (2) Each 𝜎-basis is an 𝑅-basis. Statement (1) immediately follows from the proposition, since by Theorem 6.21 (3), one has 𝐸0 = Φ1 (𝐸0 ). (2) follows from the proposition and the definition of a 𝜎-basis. Remark 10.6. (1) If Φ1 (𝐸) is complete in 𝐸 and MΦ (𝐸) is 𝜎-compact in M(𝐸), then Φ2 (𝐸) is a 𝜎- and therefore also an 𝑅-basis. (2) For the existence of a 𝜎-basis in the vector lattice 𝐸, it is necessary and sufficient that in the space M(𝐸) an open, everywhere dense, locally compact and 𝜎compact subset exists. If, for example, 𝐺 is such a subset of M with the properties mentioned, then the ideal 𝐸𝐺 = {𝑥 ∈ Φ1 (𝐸) : suppM (𝑥) ⊂ 𝐺}

(10.1)

is a 𝜎-basis in 𝐸. (3) There are Φ-bases that are not 𝑅-bases. The corresponding Counterexample 10.7 goes back to G. Ya. Lozanovskij; see [92]. Proof. (1) Due to the Corollary 6.19, the ideal Φ2 (𝐸) is complete in 𝐸 together with Φ1 (𝐸). Therefore it is only left to prove that Φ2 (𝐸) is of type (Σ). The latter is true, since according to Corollary 6.20, it is equivalent to the proposed 𝜎-compactness of MΦ .

2 Each sequence which satisfies in 𝐸0 the condition (Σ󸀠 ) is such a collection.

194 | 10 Representations of vector lattices by means of bases of finite elements (2) Let 𝐸0 be a 𝜎-basis in 𝐸. Then the set 𝐺(𝐸0 ) = ⋃𝑥∈𝐸0 𝐺𝑥 is open and locally compact, due to 𝐸0 ⊂ Φ2 (𝐸). In view of Corollary 6.19 and Theorem 6.21 (5), the set 𝐺(𝐸0 ) is 𝜎-compact and everywhere dense in M. Conversely, if 𝐺 is a subset of M(𝐸) with the properties mentioned in (2), then 𝐺 ⊂ MΦ . That the set defined in (10.1) can be taken for the required 𝜎-basis has been demonstrated in the remark following Theorem 6.21. For (3) see the next example. Example 10.7. Not every Φ-basis is an 𝑅-basis. Let (𝑁𝑡 )𝑡∈[0,1] be a system of infinite subsets of ℕ such that for 𝑡, 𝑡󸀠 ∈ [0, 1], 𝑡 ≠ 𝑡󸀠 the intersection 𝑁𝑡 ∩ 𝑁𝑡󸀠 is finite; see [87, Problem 1.6]. Denote by (𝑒󸀠𝑡 )𝑡∈[0,1] the set of all possible (infinite) sequences of positive numbers and construct the sequences (𝑒𝑡 )𝑡∈[0,1] by means of the rule 𝑒𝑡 (𝑘) = 𝑒󸀠𝑡 (𝑘)𝜒𝑁 (𝑘) for 𝑘 ∈ ℕ, 𝑡

where 𝜒𝑁 denotes the characteristic function of the set 𝑁𝑡 for 𝑡 ∈ [0, 1]. Consider now in 𝑡 the Dedekind complete vector lattice s of all real sequences the set 𝐸0 = {𝑥 ∈ s : ∃ 𝑛 ∈ ℕ, 𝜆 > 0, 𝑡1 , . . . , 𝑡𝑛 ∈ [0, 1] such that |𝑥| ≤ 𝜆(𝑒𝑡1 ∨ ⋅ ⋅ ⋅ ∨ 𝑒𝑡𝑛 )}. It is not hard to prove that 𝐸0 is a Dedekind complete vector lattice which consists of finite elements³ of s. The latter will be clear if we show that the supremum of finite many, say two, elements of kind 𝑒𝑡 belongs to c00. Indeed, consider for 𝑡 ≠ 𝑠 the coordinates of the element 𝑒𝑡 ∨ 𝑒𝑠 , i. e., (𝑒𝑡 ∨ 𝑒𝑠 )(𝑘) = 𝑒󸀠𝑡 (𝑘) 𝜒𝑁 (𝑘) ∨ 𝑒󸀠𝑠 (𝑘) 𝜒𝑁 = { 𝑡

𝑠

𝑒󸀠𝑡 (𝑘) ∨ 𝑒󸀠𝑠 (𝑘), 0,

for for

𝑘 ∈ 𝑁𝑡 ∩ 𝑁𝑠 , 𝑘 ∉ 𝑁𝑡 ∩ 𝑁𝑠

which shows that only finite many coordinates may be nonzero. Hence 𝐸0 is a Φ-basis in itself. However, 𝐸0 is not embeddable into any vector lattice of bounded elements. By way of contradiction, assume that this would be the case, i. e., 𝐸0 is embeddable into the vector lattice 𝑌 of bounded elements, where 𝑌 is supposed to be a Dedekind complete vector lattice with the order unit 𝑒. Since s is the maximal extension (see [120, § V.6]) of the vector lattice 𝐸0 , we may assume 𝑌 ⊂ s. So for each 𝑡 there is a 𝜆 𝑡 with 𝑒𝑡 ≤ 𝜆 𝑡 𝑒, i. e., 𝑒𝑡 (𝑘) ≤ 𝜆 𝑡 𝑒(𝑘) for 𝑘 ∈ ℕ. However, for the element 𝑒𝑡∗ with 𝑒󸀠𝑡∗ (𝑘) = 𝑘 𝑒(𝑘) this is impossible. This shows that 𝐸0 cannot be an 𝑅-basis. The example also shows that a vector lattice consisting of finite elements is not uniquely defined by its space of maximal ideals in general. In particular, the Dedekind complete vector lattice 𝐸0 and its Dedekind complete vector sublattice of all bounded sequences of 𝐸0 have a common space of maximal ideals, although they are not isomorphic; cf. Corollary 10.12.

3 Φ1 (s) = c00 ; see Example 3.48.

10.2 Representations by means of R-bases of finite elements

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195

For a vector lattice 𝐸 of type (𝐶𝑀 ), the ideal Φ1 (𝐸) is a 𝜎-, and consequently an 𝑅basis. Indeed, from Theorem 9.19 (1) and Remark 9.20 (1), we know that the ideal Φ1 (𝐸) which coincides with Φ2 (𝐸) (due to Corollary 6.16) is even a 𝜎-basis of finite elements.

10.2 Representations by means of R-bases of finite elements In this section we deal with a special class of representations of vector lattices, namely with those representations where a given 𝑅-basis of finite elements is represented as a vector lattice of finite functions which satisfies the condition (𝛼). These representations may be treated as a generalization of Φ𝛼-representations, which were studied in detail in Chapter 9. Definition 10.8. A representation (𝐸(𝑄), 𝑖) of 𝐸 is called a representation by means of the R-basis 𝐸0 if 𝑖(𝐸0 ) ⊂ K(𝑄) and 𝑖(𝐸0 ) satisfies the condition (𝛼). If the 𝑅-basis in 𝐸 is understood, then such a representation will be also called Rrepresentation of 𝐸. Remark 10.9. (1) The Theorems 10.1 and 8.13 imply that an R-representation of a vector lattice 𝐸 by means of an 𝑅-basis 𝐸0 is a representation on the locally compact space 𝐺(𝐸0 ) = M(𝐸0 ); cf. Remark 9.2. If 𝐸0 is a 𝜎-basis, then in addition 𝐺(𝐸0 ) is 𝜎-compact; see Remark 10.6 (2). (2) Let 𝐸 be a vector lattice with an 𝑅-basis 𝐸0 . If (𝐸(𝑆), 𝑖) and (𝐸(𝑇), 𝑗) are two Rrepresentation by means of the 𝑅-basis 𝐸0 , then a homeomorphism 𝜋 : 𝑇 → 𝑆 and some positive continuous function 𝑎 exist such that the map 𝑥(𝑠) 󳨃→ 𝑎(𝑡) (𝑥(𝜋(𝑡))) is a vector lattice isomorphism from 𝐸(𝑆) onto 𝐸(𝑇). Moreover, each isomorphism of 𝐸(𝑆) onto 𝐸(𝑇), which maps finite functions into finite functions, has such a form; see Theorems 8.13 (2) and 8.27. We prove the fundamental result on the existence of representations by means of 𝑅bases; see [92, Theorem 2]. Theorem 10.10 (Existence of basis representations). Let 𝐸 be a vector lattice and 𝐸0 an 𝑅-basis in 𝐸. Then a representation exists by means of 𝐸0 . If (𝐸(𝑄), 𝑖) is this representation, then 𝑖(𝐸0 ) = K(𝑄) if and only if 𝐸0 is uniformly complete. Proof. We denote by 𝑌 the vector lattice in which the 𝑅-basis 𝐸0 is isomorphically em̃ beddable, and by 𝑢 an arbitrary order unit in 𝑌. For each 𝑥 ∈ 𝐸0 denote by 𝑥(𝑀) = 𝑓(𝑥)

196 | 10 Representations of vector lattices by means of bases of finite elements the conditional representation with respect to 𝑢 (see Definition 6.7), where 𝑀 = 𝑓−1 (0),

where 𝑀 ∈ M(𝑌) = {𝑀󸀠 ∈ M(𝑌) : 𝑢 ∉ 𝑀󸀠 },

𝑓(𝑢) = 1.

All functions obtained in this way are continuous on M(𝑌) by Proposition 6.9. According to Remark 10.3, the spaces 𝐺𝐸 (𝐸0 ), 𝐺𝑌 (𝐸0 ) and M(𝐸0 ) are homeomorphic. We identify and denote them by 𝑄. By 𝐸0 (𝑄) denote the restrictions on 𝑄 of all conditional representations (with respect to 𝑢) of the elements of 𝐸0 and by 𝑖0 the map 𝑥 󳨃→ 𝑥|̃ 𝑄

for all 𝑥 ∈ 𝐸0 .

Then the pair (𝐸0 (𝑄), 𝑖0 ) is a representation of the vector lattice 𝐸0 . First, we convince ourselves that the image of each element 𝑥 ∈ 𝐸0 under the map 𝑖0 is in fact a finite function. This we conclude from suppM (𝑥) = supp(𝑖0 𝑥) ⊂ 𝑄, which holds, since 𝐺𝐸 (𝐸0 ) and M(𝐸0 ) are homeomorphic. So 𝑖0 𝑥 is a finite function on 𝑄. The vector lattice 𝐸0 (𝑄) satisfies the condition (𝛼). Indeed, for any ordered pair of different points 𝑀0 , 𝑀1 ∈ 𝑄 there is some 𝑥 ∈ 𝐸 with 𝑥 ∈ 𝑀0 \ 𝑀1 , which implies 1 ̃ 0 ) = 0 and 𝑥(𝑀 ̃ 1 ) ≠ 0. Then the function 𝑥(𝑀 𝑥̃ satisfies the condition of Defini𝑥(𝑀 ̃ 1) tion 8.1. The just obtained representation (𝐸0 (𝑄), 𝑖0 ) we further use for the construction of a representation of the whole vector lattice 𝐸. We now demonstrate how the isomorphism 𝑖0 can be extended to an isomorphism 𝑖 between 𝐸 and some vector lattice of continuous functions on 𝑄. Take an arbitrary positive element 𝑎 ∈ 𝐸 and put 𝑖 𝑎 = sup {𝑖0 𝑥}, 𝑥∈𝐵𝑎

where 𝐵𝑎 = {𝑥 ∈ 𝐸0 : 0 ≤ 𝑥 ≤ 𝑎}.

Observe that in fact the element 𝑎 can be represented as 𝑎 = sup𝑥∈𝐵𝑎 𝑥, because 𝐸0 is a fundament in 𝐸; see [122, Lemma VIII.4.2]. We claim that the function 𝑖 𝑎 is continuous on 𝑄. For that it suffices to show that 𝑖 𝑎 is continuous on any compact subset 𝐾 of 𝑄. Since 𝐸0 (𝑄) satisfies the condition (𝛼), according to Proposition 8.4 an element 𝑧 ∈ 𝐸0 ̃ ≥ 1 for 𝑠 ∈ 𝐾. The elements 𝑎 ∧ 𝑛 𝑧 belong to 𝐸0 for any 𝑛 ∈ ℕ. Due exists such that 𝑧(𝑠) to the equalities sup (𝑥 ∧ 𝑛 𝑧) = sup (𝑥) ∧ 𝑛 𝑧 = 𝑎 ∧ 𝑛 𝑧, 𝑥∈𝐵𝑎

𝑥∈𝐵𝑎

we also get sup 𝑖0 (𝑥 ∧ 𝑛 𝑧) = 𝑖0 (𝑎 ∧ 𝑛 𝑧).

𝑥∈𝐵𝑎

Since the infimum of the function 𝑧̃ on 𝐾 is positive, for sufficiently large 𝑛 we have 𝑖0 (𝑥 ∧ 𝑛 𝑧)(𝑡) = 𝑖0 𝑥(𝑡),

𝑡 ∈ 𝐾.

On 𝐾 a continuous function is now constructed which apparently does not depend on the choice of the element 𝑧. Due to the fact that 𝐾 was an arbitrary compact subset

10.2 Representations by means of R-bases of finite elements

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of 𝑄, some function 𝑎(𝑡) is continuously defined on the whole of the space 𝑄. The assignment 𝑎 󳨃→ 𝑎(𝑡) for nonnegative elements of 𝐸 is additive and isotone, and therefore allows a unique extension on the whole vector lattice 𝐸 to some linear monotone map 𝑖 : 𝐸 → 𝐶(𝑄); see [120, Lemma VIII.1.1]. It is easy to see that this map is the required isomorphism between 𝐸 and some vector lattice 𝐸(𝑄) of continuous functions on 𝑄. It is also clear that the pair (𝐸(𝑄), 𝑖) is a representation by means of the 𝑅-basis 𝐸0 . The last statement of the theorem follows from Theorem 8.13 (3). Remark 10.11. If the vector lattice 𝐸 itself is an 𝑅-basis then a representation 𝐸(𝑄) exists with the following properties: (1) 𝐸(𝑄) consists of finite functions; (2) 𝐸(𝑄) satisfies the condition (𝛼); (3) 𝑄 is locally compact and homeomorphic to M(𝐸). If, moreover, 𝐸 is vector lattice of type (Σ), then 𝑄 is also 𝜎-compact. We now get the following important fact. Corollary 10.12. If a uniformly complete vector lattice 𝐸 consists only of finite elements and is embeddable into a vector lattice of bounded elements, then 𝐸 is uniquely defined by its space of maximal ideals M(𝐸). Indeed, according to the previous remark a representation 𝐸(𝑄) exists on the space 𝑄 = M(𝐸). Due to the uniform completeness, the collection of all finite continuous functions on M(𝐸) is isomorphic to 𝐸(𝑄), and consequently isomorphic to the original vector lattice 𝐸; compare this corollary to the remark following Example 10.7. Corollary 10.13. If 𝐸 = Φ1 (𝐸) and 𝐸 is a vector lattice of type (Σ), then an 𝛼representation exists as a vector lattice of finite functions on a locally compact 𝜎compact space; see Theorem 9.15. According to the assumptions, the vector lattice itself is a 𝜎-, and therefore an 𝑅-basis. The corollary follows from Remark 10.11. We now demonstrate that the completeness of Φ2 (𝐸) in a vector lattice 𝐸, i. e., 𝑥 = 0 whenever 𝑥 ∈ 𝐸 and 𝑥 ⊥ Φ2 (𝐸), is sufficient for the existence of an 𝑅-basis. Theorem 10.14. If the ideal Φ2 (𝐸) is complete in 𝐸, then an 𝑅-basis exists in 𝐸. Proof. In any vector lattice a complete set of pairwise disjoint positive elements exists; see [120, Lemma IV.7.1]. Denote by (𝑒𝜉 )𝜉∈Ξ such a system in the ideal Φ2 (𝐸) and introduce the following vector lattices of bounded elements: 𝐸𝜉 = { 𝑥 ∈ 𝐸 : ∃ 𝜆 > 0;

|𝑥| ≤ 𝜆𝑒𝜉 }.

Let 𝐸0 = { 𝑥 ∈ 𝐸 : 𝑥 = ∑𝑛𝑘=1 𝑥𝑘 ; 𝑥𝑘 ∈ 𝐸𝜉𝑘 , 𝑛 = 𝑛(𝑥)}. It should be clear that 𝐸0 consists of totally finite elements and is a complete set in 𝐸. In order to establish that 𝐸0 is an 𝑅-basis, it suffices to prove that 𝐸0 is embeddable into a vector lattice of bounded elements. According to Theorem 2.15 (S. Kakutani, H.F. Bohnenblust, M.G. Krein, S.G. Krein), the vector lattice 𝐸𝜉 is Riesz isomorphic to some vector sublattice of the

198 | 10 Representations of vector lattices by means of bases of finite elements vector lattice 𝐶(𝑄𝜉) of all continuous functions on some compact topological space 𝑄𝜉 , where the image of the element 𝑒𝜉 may be assumed to be the function identical to 1 on 𝑄𝜉 for all 𝜉 ∈ Ξ. If the direct sum of the compact spaces 𝑄𝜉 is denoted by 𝑄, then it is easy to see that 𝐸0 is embeddable into the vector lattice of all bounded continuous functions on 𝑄. Theorem 10.15. For the existence of a 𝜎-basis of finite elements in a vector lattice 𝐸 it is necessary and sufficient that a sequence (𝑥𝑛)𝑛∈ℕ of elements exists in 𝐸 with the following properties: (1) 0 ≤ 𝑥1 ≤ 𝑥2 ≤ . . . ; (2) (𝑥𝑛+2 − 𝑥𝑛+1 ) ⊥ 𝑥𝑛 , 𝑛 ∈ ℕ ; (3) for each 𝑥 ∈ 𝐸 there is a number 𝜆 > 0 with 𝜆 |𝑥| = sup𝑛 (𝜆 |𝑥| ∧ 𝑥𝑛 ). Proof. Corollary 10.5 (2) and Theorem 10.10 guarantee the existence of an R-representation by means of the given 𝜎-basis on some 𝜎-compact space. The construction of a sequence with the corresponding properties is straightforward and makes use of Proposition 8.4. Sufficiency. Put 𝐸0 = {𝑥 ∈ 𝐸 : ∃ 𝑐, 𝑛;

|𝑥| ≤ 𝑐𝑥𝑛}

and notice that 𝐸0 turns out to be a vector lattice of type (Σ) which is solid and complete in 𝐸. For 𝐸0 to be a 𝜎-basis it has to be shown that each element in 𝐸0 is totally finite in 𝐸. So we will show that for any index 𝑛 ∈ ℕ and for each 𝑥 ∈ 𝐸 there is a real 𝑐𝑥 > 0 such that the inequality |𝑥| ∧ 𝑘𝑥𝑛 ≤ 𝑐𝑥 𝑥𝑛+1

for all 𝑘 ∈ ℕ

(10.2)

is true. Fix 0 ≤ 𝑥 ∈ 𝐸 and 𝑛 ∈ ℕ. Let 𝜆 be the real which exists for 𝑥 according to condition (3). Without loss of generality 𝜆 ≤ 1, may be assumed. Then from 𝜆 𝑥 = sup𝑛 (𝜆 𝑥 ∧ 𝑥𝑛) we have for any 𝑘 ∈ ℕ 𝑥 ∧ 𝑘 𝑥𝑛 ≤

1 1 (𝜆 𝑥 ∧ 𝑘 𝑥𝑛 ) = sup(𝜆 𝑥 ∧ 𝑘 𝑥𝑛 ∧ 𝑥𝑙 ), 𝜆 𝜆 𝑙

and for each 𝑙 ≥ 𝑛 + 2 (due to 0 = (𝑥𝑙 − 𝑥𝑙−1 ) ∧ 𝑥𝑙−2 ≥ (𝑥𝑙 − 𝑥𝑙−1 ) ∧ 𝑥𝑛 = 0), the estimation 𝑥𝑙 ∧ 𝑘 𝑥𝑛 = (𝑥𝑙−1 + (𝑥𝑙 − 𝑥𝑙−1 )) ∧ 𝑘 𝑥𝑛 = 𝑥𝑙−1 ∧ 𝑘 𝑥𝑛 ≤ 𝑥𝑛+1 ∧ 𝑘 𝑥𝑛 . From there the relations sup𝑙≥𝑛+2 (𝜆 𝑥 ∧ 𝑥𝑙 ∧ 𝑘 𝑥𝑛 ) ≤ 𝜆𝑥 ∧ 𝑥𝑛+1 ∧ 𝑘𝑥𝑛 and 𝑥 ∧ 𝑘 𝑥𝑛 ≤

1 1 ( sup (𝜆 𝑥 ∧ 𝑘 𝑥𝑛 ∧ 𝑥𝑙 ) ∨ sup (𝜆 𝑥 ∧ 𝑘 𝑥𝑛 ∧ 𝑥𝑛+1 )) ≤ 𝑥𝑛+1 𝜆 𝑙≤𝑛+1 𝜆 𝑙≥𝑛+2

immediately follow. With 𝑐𝑥 =

1 𝜆

we get (10.2).

Now we are able to demonstrate the connection between the existence of a Φ𝛼representation of a vector lattice and some topological properties of the space of its maximal ideals. The next theorem is analogous to Theorem 9.12.

10.3 Some properties of the realization space

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Theorem 10.16. For the existence of a Φ𝛼-representation (𝐸(𝑄), 𝑖) for a vector lattice 𝐸 on a 𝜎-compact space 𝑄 it is sufficient and, if 𝐸 is of type (Σ) and uniformly complete, also necessary, that the space M(𝐸) is locally compact and 𝜎-compact. Proof. The proof of the necessity is the same as for Theorem 9.12; see Remark 9.13 (3). Sufficiency. The local compactness and 𝜎-compactness of M(𝐸) imply (by Remark 6.14 (2)) the relation M = MΦ such that MΦ (apart from its local compactness), is 𝜎-compact and therefore satisfies the conditions of Remark 10.6 (2). Hence Φ1 (𝐸) is a 𝜎- and, it follows, also an 𝑅-basis in 𝐸. It is clear that the corresponding R-representation of 𝐸, by means of the basis Φ1 (𝐸), is a Φ𝛼-representation on the 𝜎-compact space 𝐺(Φ1 (𝐸)) = MΦ .

10.3 Some properties of the realization space In the previous two sections methods have been elaborated to describe representations of vector lattices by means of 𝑅- or 𝜎-bases. For a representation (𝐸(𝑄), 𝑖) of a vector lattice 𝐸 by means of an 𝑅-basis 𝐸0 , e. g., in Theorem 10.10, the question was answered when the vector lattice 𝐸(𝑄) contains all finite functions on 𝑄. In this case, the uniform completeness of the 𝑅-basis was a necessary and sufficient condition. In the present section, we first deal with the normality⁴ of the isomorphic embedding of 𝐸(𝑄) into 𝐶(𝑄), and later investigate the connectedness, the local connectedness, and the metrizability of the topological space 𝑄. It is reasonable to provide (without proof) some results on convergent nets in 𝐶(𝑄) for a locally compact space 𝑄, where for simplicity we restrict the formulations only to sequences of continuous functions. Proposition 10.17. Let 𝐶(𝑄) be the space of all real continuous functions on the locally compact topological space 𝑄 and (𝑥𝑛)𝑛∈ℕ a sequence in 𝐶(𝑄). Then the following statements hold: (1) if the sequence (𝑥𝑛 )𝑛∈ℕ converges compactly on 𝑄 to some function 𝑥0 , i. e., for ar󵄨 󵄨 bitrary 𝜀 > 0, 𝑛 ∈ ℕ exists such that 󵄨󵄨󵄨𝑥𝑛(𝑡) − 𝑥0 (𝑡)󵄨󵄨󵄨 ≤ 𝜀 on any compact subset of 𝑄, then 𝑥0 ∈ 𝐶(𝑄); (2) if the sequence (𝑥𝑛)𝑛∈ℕ is compactly preconvergent (compactly Cauchy) on 𝑄, i. e., 󵄨 󵄨 for arbitrary 𝜀 > 0, 𝑛 ∈ ℕ exists such that for any 𝑝 ∈ ℕ one has 󵄨󵄨󵄨󵄨𝑥𝑛 (𝑡) − 𝑥𝑛+𝑝 (𝑡)󵄨󵄨󵄨󵄨 ≤ 𝜀 on any compact subset of 𝑄, then it converges compactly on 𝑄; (3) if the sequence (𝑥𝑛 )𝑛∈ℕ is uniformly convergent in 𝐶(𝑄) (uniformly Cauchy), then it is compactly convergent (compactly preconvergent). Definition 10.18. A uniformly complete 𝑅-basis 𝐸0 of the vector lattice 𝐸 is called normal if any family (𝜉𝛼 )𝛼∈𝐴 of nonnegative elements of 𝐸0 with the properties: 󵄨 󵄨 4 This means 𝐸(𝑄) is a solid subset of 𝐶(𝑄), i. e., 𝑦 ∈ 𝐶(𝑄), 𝑥 ∈ 𝐸(𝑄) and 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ≤ |𝑥| imply 𝑦 ∈ 𝐸(𝑄); see p. 8.

200 | 10 Representations of vector lattices by means of bases of finite elements (a) for each element 𝑧 ∈ 𝐸0 a (finite) index set⁵ 𝐴 𝑧 ∈ F(𝐴) exists such that 𝜉𝛼 ⊥ 𝑧 for all 𝛼 ∉ 𝐴 𝑧 ; (b) all finite subfamilies of (𝜉𝛼 )𝛼∈𝐴 are bounded in 𝐸, i. e., an element 𝑢 ∈ 𝐸 exists such that ∑𝛼∈𝐹 𝜉𝛼 ≤ 𝑢 for any subset 𝐹 ∈ F(𝐴); is order summable in 𝐸, i. e., (𝑜)- lim ∑𝛼∈𝐹 𝜉𝛼 exists in 𝐸. 𝐹∈F(𝐴)

Theorem 10.19. Let (𝐸(𝑄), 𝑖) be a representation of a vector lattice 𝐸 by means of a uniformly complete 𝑅-basis 𝐸0 . For the normality of embedding 𝐸(𝑄) into 𝐶(𝑄) it is necessary, and if 𝑄 is 𝜎-compact, also sufficient, that the basis 𝐸0 is normal. Proof. Necessity. Consider a family (𝜉𝛼 )𝛼∈𝐴 of elements of 𝐸0 with the properties listed in the previous definition, and let 𝐾 be an arbitrary compact subset of 𝑄. Then by Proposition 8.4 an element 𝑥0 ∈ 𝐸0 exists such that 𝐾 ⊂ {𝑡 ∈ 𝑄 : 𝑥0 (𝑡) ≠ 0} and a finite index set 𝐴 0 ⊂ 𝐴 with the property that for all 𝛼 ∉ 𝐴 0 the functions 𝑖(𝜉𝛼 ) vanish at 𝐾. Therefore the net 𝑖(𝜉𝛼 )𝛼∈𝐴 compactly converges on 𝐾 to some continuous function 𝑎. If 𝑢 ∈ 𝐸 denotes a common boundary for all finite subfamilies (𝜉𝛼 )𝛼∈𝐹 with 𝐹 ∈ F(𝐴) (the existence of 𝑢 is guaranteed by the condition (b) of the previous definition), then 𝑎(𝑡) ≤ (𝑖𝑢)(𝑡) holds for all 𝑡 ∈ 𝑄. Due to the normality of 𝐸(𝑄) the element 𝑖−1 𝑎 belongs to 𝐸. By a standard argument it can be shown that 𝑖−1 𝑎 = (𝑜)- lim ∑ 𝜉𝛼 . 𝐹∈F(𝐴)

𝛼∈𝐹

This shows that the base 𝐸0 is normal. Sufficiency. As mentioned in Remark 10.9 (1), the space 𝑄 is locally compact. Due to the 𝜎-compactness, the space 𝑄 has a representation as 𝑄 = ⋃𝑛∈ℕ 𝐾𝑛 , where (𝐾𝑛)𝑛∈ℕ is a sequence of compact subsets of 𝑄 which satisfies the condition 𝐾𝑛 ⊂ int(𝐾𝑛+1 ) for each 𝑛. Let 𝑦 ∈ 𝐶(𝑄), 𝑥 ∈ 𝐸(𝑄), and 0 ≤ 𝑦(𝑡) ≤ 𝑥(𝑡), 𝑡 ∈ 𝑄. With the help of condition (𝛼) one constructs nonnegative functions 𝑥𝑛 ∈ 𝐸0 (𝑄) such that 𝑥𝑛 (𝑡) = 𝑦(𝑡) on 𝐾𝑛 and 𝑥𝑛 (𝑡) = 0 on 𝑄 \ 𝐾𝑛+1 . Moreover, 𝑥𝑛 (𝑡) ≤ 𝑦(𝑡) can be achieved on 𝑄. The elements 𝑖−1 𝑥𝑛 belong to 𝐸0 and all partial sums ∑𝑙𝑛=1 𝑖−1 𝑥𝑛 are bounded by the element 𝑖−1 𝑥. The elements 𝜉1 = 𝑖−1 𝑥1 , 𝜉2 = 𝑖−1 𝑥2 − 𝑖−1 𝑥1 , . . . , 𝜉𝑛 = 𝑖−1 𝑥𝑛 − 𝑖−1 𝑥𝑛−1 , . . . satisfy conditions (a) and (b) of Definition 10.18. Consequently an element 𝑎 ∈ 𝐸 exists with 𝑙

𝑎 = (𝑜)- lim ∑ 𝜉𝑗 = sup 𝑖−1 𝑥𝑙 . 𝑙→∞

𝑗=1

5 F(𝐴) denotes the collection of all finite subsets of 𝐴.

𝑙

10.3 Some properties of the realization space

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On each compact subset of 𝑄 the sequence 0 ≤ 𝑥1 ≤ 𝑥2 ≤ . . . , converges uniformly, on the one hand to the function 𝑖 𝑎, and on the other, due to its construction, to the function 𝑦. So we get (𝑖 𝑎)(𝑡) = 𝑦(𝑡) for all 𝑡 ∈ 𝑄, which shows that the element 𝑎 is the preimage of the function 𝑦. This means 𝐸(𝑄) is normally embedded in 𝐶(𝑄). Remark 10.20. (1) If (𝐸(𝑄), 𝑖) is a representation of 𝐸 by means of an 𝑅-basis 𝐸0 , then the normality of the basis (loosely speaking) means that the (𝑜)-summability of infinite families of finite functions in 𝑖(𝐸0 ) with the property that the supports of only finitely many of them have a common interior point, or in other words, on each compact subset of 𝑄 all functions of such a family vanish with the exception of a finite number. (2) The 𝜎-compactness of 𝑄 is guaranteed if the representation of 𝐸 happens by means of a 𝜎-basis; see Remark 10.9 (1). Not every uniformly complete 𝜎-basis is normal, as the following example shows. Example 10.21. A vector lattice of type (Σ) in which a uniform complete 𝜎-basis is not normal, although it satisfies the condition (𝜇). Similar to Example 9.21, consider the vector lattice 𝐸(𝑇) of all continuous functions on 𝑇 = [1, +∞) ⊂ ℝ, each of which is a polynomial from some real number 𝜆 ≥ 1 on. The sequence (𝑡𝜈)𝜈∈ℕ , 𝑡 ∈ 𝑇 satisfies in 𝐸(𝑇) the condition (Σ󸀠 ). The set of all finite functions of 𝐸(𝑇) is a 𝜎-basis in 𝐸(𝑇). It is clear that this 𝜎-basis is uniformly complete. However, since 𝐸(𝑇) is not normally included in 𝐶(𝑇), according to Theorem 10.19 this 𝜎-basis cannot be normal. The last fact can also be shown directly by considering the 2 + series ∑∞ 𝑛=1 (1 − (𝑡 − 2𝑛) ) , whose sum does not belong to 𝐸(𝑇). The property K(𝑄) ⊂ 𝐸(𝑄) or the normality of 𝐸(𝑄) in 𝐶(𝑄) for a representation (𝐸(𝑄), 𝑖) of 𝐸 by means of an 𝑅-basis is a property of the 𝑅-basis, as the corresponding theorems have shown, and therefore holds for any representation of 𝐸 by means of this basis. The normality of a 𝜎-basis was first used in [126, Theorem 3] to represent uniformly complete vector lattices. As already mentioned at the end of Section 10.1, in a vector lattice 𝐸 of type (𝐶𝑀 ) the set Φ1 (𝐸) turns out to be a 𝜎-basis. If, in addition, 𝐸 is uniformly complete, then Φ1 (𝐸) is even a normal basis. This is seen as follows: first, according to Theorem 10.10, a representation (𝐸(𝑄), 𝑖) exists by means of the 𝜎-basis Φ1 (𝐸), which is obviously a Φ𝛼-representation of 𝐸, where, due to uniform completeness, Theorem 9.19 implies that 𝐸(𝑄) is normal in 𝐶(𝑄). By the previous theorem the basis Φ1 (𝐸) is then normal. We now study the connectedness and the local connectedness of 𝑄, and will formulate a corresponding condition in the language of the bases; see [124, 128]. It is easy to understand that the condition should avoid the existence of projections onto nontrivial bands in the vector lattice. It is well known that projections in the vector lattice 𝐶(𝑄) for compact 𝑄 are closely related to the existence of nontrivial open-closed subset in 𝑄. Some extensive studies in this direction, although somewhat different to our main argument, are found in [119] for example.

202 | 10 Representations of vector lattices by means of bases of finite elements Definition 10.22. A positive element 𝑥 of a vector lattice 𝐸 is called decomposable, if elements 𝑥1 , 𝑥2 ∈ 𝐸 exist such that 0 ≤ 𝑥𝑖 ,

𝑖 = 1, 2,

𝑥1 ⊥ 𝑥2 ,

𝑥 = 𝑥1 + 𝑥2 .

If an element 0 ≤ 𝑥 ∈ 𝐸 is indecomposable, then it is also not decomposable in any ideal 𝑌 ⊂ 𝐸 to which it belongs. Otherwise the decomposition of 𝑥 in 𝑌 would also be a decomposition in 𝐸. Conversely, if 𝑥 is indecomposable in the ideal 𝐼𝑥 = {𝑧 ∈ 𝐸 : |𝑧| ≤ 𝜆 𝑧 𝑥}, then it is indecomposable in 𝐸 as well. Otherwise, due to the normality of 𝐼𝑥 in 𝐸, the element would have a decomposition also in 𝐼𝑥 . In the vector lattice 𝐶(𝑄) of all continuous functions on a topological space 𝑄, the indecomposable elements are characterized according to the following proposition. Proposition 10.23. Let 𝑄 be topological space and 𝑧 a nonnegative function of 𝐶(𝑄). Let 𝐺 = {𝑡 ∈ 𝑄 : 𝑧(𝑡) > 0}. The following assertions are equivalent: (1) 𝐺 is a connected subset; (2) the element 𝑧 is indecomposable in 𝐶(𝑄); (3) the element 𝑧 is indecomposable in 𝐼𝑧 . Proof. The equivalence of (2) and (3) is already clear and the implication (1) ⇒ (2) is trivial. (2) ⇒ (1). Without loss of generality we assume 𝑧 > 0. If 𝐺 is not connected then 𝐺 = 𝐺1 ∪ 𝐺2 ,

𝐺1 ∩ 𝐺2 = 0,

𝐺1 , 𝐺2 ≠ 0,

where 𝐺1 , 𝐺2 are open sets. If 𝑡0 is a boundary point of 𝐺1 , then 𝑡0 ∉ 𝐺1 and 𝑧(𝑡0 ) = 0. One has also 𝑡0 ∉ 𝐺2 , otherwise 𝑡0 would be an interior point of 𝑄 \ 𝐺1 . So 𝑡0 ∉ 𝐺. Similarly, the boundary points of 𝐺2 do not belong to 𝐺 either. The functions {𝑧(𝑡), 𝑡 ∈ 𝐺𝑖 𝑧𝑖 (𝑡) = { , 0, 𝑡 ∉ 𝐺𝑖 {

𝑖 = 1, 2

are continuous and disjoint to each other. Since both 𝑧𝑖 > 0 and 𝑧 = 𝑧1 + 𝑧2 , it follows that the element 𝑧 is decomposable. Observe that a positive element 𝑧 of a vector lattice 𝐸 is indecomposable if and only if the set 𝐺𝑧 = {𝑀 ∈ M(𝐸) : 𝑧 ∉ 𝑀} is connected. Indeed, by Proposition 6.9, the conditional representations 𝑥̃ of all elements 𝑥 ∈ 𝐼𝑧 are continuous functions on the set 𝐺𝑧 . In particular, 𝑧̃ ∈ 𝐶(𝐺𝑧 ). It remains to apply the previous proposition. Theorem 10.24. Let (𝐸(𝑄), 𝑖) be a representation of a vector lattice 𝐸 by means of an 𝑅basis 𝐸0. If in 𝐸0 a system (𝑥𝛼)𝛼∈𝐴 exists of indecomposable elements with the properties: (a) ⋃𝛼∈𝐴 𝐺𝛼 = 𝑄 , where 𝐺𝛼 = {𝑡 ∈ 𝑄 : 𝑥𝛼 (𝑡) ≠ 0}, 𝛼 ∈ 𝐴; (b) for each element 𝑥 ∈ 𝐸0 there are an index 𝛼0 and a real 𝑐𝑥 > 0 such that |𝑥| ≤ 𝑐𝑥 𝑥𝛼0 ; then the space 𝑄 is connected.

10.3 Some properties of the realization space

| 203

Proof. According to Proposition 10.23, all sets 𝐺𝛼 are connected. The sets (𝐺𝛼 )𝛼∈𝐴 create a directed system of connected sets, since for arbitrary 𝛼󸀠, 𝛼󸀠󸀠 ∈ 𝐴, due to property (b), there is an index 𝛼0 with 𝑥𝛼󸀠 + 𝑥𝛼󸀠󸀠 ≤ 𝑐 𝑥𝛼0 . This yields 𝐺𝛼󸀠 , 𝐺𝛼󸀠󸀠 ⊂ 𝐺𝛼0 . By means of property (a), Theorem 2󸀠 of [76, § 46] guarantees the connectedness of 𝑄. In order to study the question when the space 𝑄 is locally connected, we make use of a notion introduced in [140] and adapt it for our purposes; see [124]. Definition 10.25. An 𝑅-basis 𝐸0 of a vector lattice 𝐸 is called a 𝑊-basis if, for each element 0 < 𝑥 ∈ 𝐸0 and for any real 𝜀 > 0, an element 𝑦 ∈ 𝐸0 exists satisfying the properties: (a) 0 < 𝑦 ≤ 𝑥; (b) 𝑦 = 𝑦1 +⋅ ⋅ ⋅+𝑦𝑁 for 𝑁 = 𝑁(𝑥) with 𝑦𝑖 ⊥ 𝑦𝑗 for 𝑖 ≠ 𝑗 and 𝑦𝑗 are indecomposable in 𝐸0 for 𝑗 = 1, . . . , 𝑁; 󵄨 󵄨 (c) 󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 ≤ 𝜀𝑢, where 𝑢 ∈ 𝐸0 is a majorant of the totally finite element 𝑥; cf. Theorem 6.21(3). Theorem 10.26. Let (𝐸(𝑄), 𝑖) be a representation of the vector lattice 𝐸 by means of the uniformly complete 𝑅-basis 𝐸0 . Then for 𝑄 to be locally connected it is necessary and sufficient that 𝐸0 is a 𝑊-basis. Proof. Necessity. By Theorem 10.10 one has 𝐸0 (𝑄) = 𝑖(𝐸0 ) = K(𝑄), i. e., 𝐸0 (𝑄) is the set of all finite functions on 𝑄. The image of 𝑥 ∈ 𝐸 under the Riesz isomorphism 𝑖 is denoted by 𝑥.̃ For 𝐸0 to be a 𝑊-basis we fix an arbitrary positive element 𝑥 ∈ 𝐸0 together with some of its majorants 𝑢 ∈ 𝐸0 and arbitrary 𝜀 > 0. The system 𝑘(𝐺) of ̃ ≠ 0} consists of disjoint all connected components of the open set 𝐺 = {𝑡 ∈ 𝑄 : 𝑥(𝑡) open (and therefore open-closed) sets (see [31, p. 175], [76, § 49.II, Theorem 4]) and ̃ therefore is an open covering for the compact set 𝐹 = {𝑡 ∈ 𝑄 : 𝑥(𝑡) ≥ 𝜀}. Denote by 𝐺1 , 𝐺2 , . . . , 𝐺𝑁 any finite subcovering of 𝑘(𝐺) for 𝐹. Due to the properties of the system {𝐺1 , 𝐺2 , . . . , 𝐺𝑁 }, the functions {𝑥(𝑡), ̃ 𝑡 ∈ 𝐺𝑗 , 𝑦𝑗 (𝑡) = { 0, 𝑡 ∈ 𝑄 \ 𝐺𝑗 {

𝑗 = 1, 2, . . . , 𝑁

are continuous, not identically zero and pairwise disjoint. Since 𝐺𝑗 is connected and ̃ all elements 𝑦𝑗 are indecomposable in the vector lies in the compact set 𝐾 = supp(𝑥), lattice 𝐸0 (𝑄) for all 𝑗 = 1, . . . , 𝑁. The function 𝑁

{ { 0, 𝑡 ∈ ⋃ 𝐺𝑗 { { 𝑗=1 ̃ − ∑ 𝑦𝑗 (𝑡) = { 𝑎(𝑡) = 𝑥(𝑡) 𝑁 { { 𝑗=1 {𝑥(𝑡), ̃ 𝑡 ∈ 𝑄 \ ⋃ 𝐺𝑗 𝑗=1 { 𝑁

204 | 10 Representations of vector lattices by means of bases of finite elements can be nonzero at most on 𝐾, where 𝑢(𝑡) ≥ 𝛼0 > 0 holds for some 𝛼0 ; cf. Theorem 6.13. In view of 𝐹 ⊂ ⋃𝑁 𝑗=1 𝐺𝑗 , one has the estimates max 𝑎(𝑠) = 𝑠∈𝐾

̃ < 𝜀 ≤ 𝜀 max 𝑥(𝑠) 𝑁

𝑠∈𝐾\ ⋃ 𝐺𝑗

1 𝑢(𝑡) 𝛼0

for 𝑡 ∈ 𝐾.

𝑗=1

Outside 𝐾 the function 𝑎 vanishes, which leads to 𝑁

|𝑥 − ∑ 𝑦𝑗 | < 𝑗=1

𝜀 𝑢. 𝛼0

Sufficiency. We have to show that any point of 𝑄 possesses a neighborhood basis which consists of connected sets. Let 𝑡0 ∈ 𝑄 be an arbitrary point and 𝑈 an open neighborhood of 𝑡0 . Due to the set 𝐸0 (𝑄) satisfying the condition (𝛼), there is a positive function 𝑥0 ∈ 𝐸0 (𝑄) with 𝑥0 (𝑡0 ) > 0, and 𝑥0 (𝑡) = 0 for 𝑡 ∈ 𝑄 \ 𝑈; see Corollary 8.7. It is clear that 𝑢(𝑡0 ) > 0 holds also for any majorant 𝑢 ∈ 𝐸0 of the finite element 𝑥0 . For the positive 𝑥0 (𝑡0 ) number 𝜀 = 2𝑢(𝑡 , by assumption in 𝐸0 a positive element 𝑦 = 𝑦1 + ⋅ ⋅ ⋅ 𝑦𝑁 exists with 0) the properties formulated in the definition of a 𝑊-basis, in particular, 0 < 𝑦 ≤ 𝑥0 . The function 𝑦 cannot vanish at the point 𝑡0 , otherwise we would get 󵄨󵄨󵄨𝑥0 (𝑡0 ) − 𝑦(𝑡0 )󵄨󵄨󵄨 = 𝑥0 (𝑡0 ) = 2𝜀𝑢(𝑡0 ) > 𝜀𝑢(𝑡0 ), 󵄨 󵄨 which is in contradiction to the property (c) of a 𝑊-basis. So 𝑦(𝑡0 ) > 0 implies that 𝑦𝑘0 (𝑡0 ) > 0 for some 𝑘0 ∈ {1, . . . , 𝑁}. Hence the open set 𝐺 = {𝑡 ∈ 𝑄 : 𝑦𝑘0 (𝑡) > 0} is connected, since the element 𝑦𝑘0 is indecomposable. Because the function 𝑦𝑘0 is continuous, the set 𝐺 is a neighborhood of 𝑡0 which is contained in 𝑈. From the proved theorem one automatically also obtains conditions for the local connectedness of the spaces 𝐺(𝐸0 ) and M(𝐸0 ), since in the present situation both are homeomorphic to 𝑄. If, however, Proposition 6.4 is taken into consideration, then (less trivially) a condition for the local connectedness even of the space M(𝐸) can be obtained, namely: for a uniformly complete vector lattice 𝐸, the space M(𝐸) is locally connected if and only if a uniformly complete 𝑊-basis exists in 𝐸. Now we are interested in the metrizability of the space 𝑄. For that the following definition is necessary. Definition 10.27. A vector lattice 𝐸 is called (𝑟)-separable if a countable (𝑟)-dense subset exists in 𝐸; see also Definition 9.22. Theorem 10.28. Let 𝐸 be a vector lattice, 0 < 𝑧 ∈ 𝐸, and (𝐸(𝑄), 𝑖) a representation of 𝐸 where the points of the set 𝐺 = {𝑡 ∈ 𝑄 : 𝑧(𝑡) > 0} are strongly separated by 𝐸(𝑄). If the ideal 𝐼𝑧 = {𝑥 ∈ 𝐸 : |𝑥| ≤ 𝜆 𝑥 𝑧} is (r)-separable in 𝐸, then any compact subset of 𝐺 is metrizable. Proof. Let 𝐾 be an arbitrary fixed compact subset of 𝐺. Denote by 𝑌 the vector lattice of the restrictions of all functions of 𝑖(𝐼𝑧 ) onto 𝐾. The space 𝑌, equipped with the max󵄩 󵄩 imum norm 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 = max𝑡∈𝐾 𝑦(𝑡) inherits the separability from the (𝑟)-separabilty of 𝐼𝑧 ,

10.3 Some properties of the realization space

|

205

since the (𝑟)-convergence in 𝐸(𝑄) induces the uniform convergence on 𝐾. The vector lattice 𝑌 strongly separates the points of 𝐾. Therefore according to the Stone–Weierstrass Theorem (see [107, Theorem V.8.1]), the uniform closure of 𝑌 is 𝐶(𝐾). Hence 𝐶(𝐾) is separable, which yields the metrizability of the compact set 𝐾; see [123]. The theorem has the following corollaries. Corollary 10.29. (1) Let 𝐸 be a vector lattice and 𝑍 a vector sublattice of 𝐸 which satisfies the condition: each order interval 𝐼𝑧 = {𝑥 ∈ 𝐸 : |𝑥| ≤ 𝜆 𝑥𝑧} for 𝑧 ∈ 𝑍 is (r)-separable. If (𝐸(𝑄), 𝑖) is a representation of 𝐸, where the set 𝑖(𝑍) strongly separates the points of 𝑄, then each compact subset 𝐾 of the space 𝑄 is metrizable. This corollary implies the metrizability of the space 𝑄. (2) Let the conditions of part (1) of the corollary be satisfied and let the space 𝑄 be locally compact and 𝜎-compact. Then 𝑄 is metrizable. For the proof we mention that, according to the previous part, the space 𝑄 is locally metrizable, i. e., each point of 𝑄 possesses a neighborhood which in turn is a metrizable space. Since a locally compact, 𝜎-compact space is always paracompact (see [31, p. 141, Theorem V]), the metrizability of the space 𝑄 is a consequence of a Theorem of Yu. M. Smirnov; see [113, Theorem 3]. Now we are able to characterize the metrizability of 𝑄 by means of 𝜎-bases. (3) Let (𝐸(𝑄), 𝑖) be a representation of the vector lattice 𝐸 by means of a 𝜎-basis 𝐸0 . For the metrizability of the space 𝑄 it is necessary and sufficient that each order ideal 𝐼𝑧 for 𝑧 ∈ 𝐸0 is (𝑟)-separable. Sufficiency. Due to Remark 10.9 (1), the space 𝑄 is locally compact and 𝜎-compact. The image 𝐸0 (𝑄) under 𝑖 of the vector lattice 𝐸0 consists of finite functions and satisfies the condition (𝛼). The statement immediately follows from (2). Necessity. If 𝑄 is metrizable, then any compact subset of 𝑄 is metrizable as well. In particular, the set supp(𝑖 𝑧) is metrizable for each element 𝑧 ∈ 𝐸0 . Due to condition (𝛼) for an arbitrary element 𝑧 ∈ 𝐸0 , there is a finite function in 𝑦 ∈ 𝐸0 (𝑄) such that 𝑦(𝑡) ≥ 1

for

𝑡 ∈ supp(𝑖 𝑧).

The separability of the space 𝐶(supp(𝑦)) with the maximum norm implies the (𝑟)separability of 𝐼𝑧 . (4) Let 𝐸 be an (𝑟)-complete vector lattice of type (Σ) with Φ1 (𝐸) = Φ2 (𝐸), in which the ideal Φ2 (𝐸) is a vector lattice of type (Σ). If Φ1 (𝐸) is complete in 𝐸, then the space (M(𝐸), 𝜏ℎ𝑘 ) is metrizable if and only if each order ideal 𝐼𝜑 for 𝜑 ∈ Φ1 (𝐸) is (𝑟)-separable. Indeed, according to Corollary 6.33, under the given assumptions the space (M(𝐸), 𝜏ℎ𝑘 ) is locally compact and 𝜎-compact. The claim directly follows from part (3) of the corollary, if for 𝐸 there exists a representation on some space 𝑄 which is isomorphic to M(𝐸). The latter is guaranteed by Theorem 10.16.

206 | 10 Representations of vector lattices by means of bases of finite elements Our considerations might thus be summed up: the topological space M(𝐸) of an (𝑟)complete vector lattice 𝐸 (under the assumption of (𝑟)-separability of its order ideals 𝐼𝜑 for 𝜑 ∈ Φ1 (𝐸)) is metrizable if M(𝐸) is locally compact and 𝜎-compact, and if 𝐸 possesses a representation on a space that is homeomorphic to M(𝐸); see Proposition 9.10 and Theorem 9.12.

—– ∘ ∘ ∘ —–

List of Examples The list includes several examples which are dealt with in the book. They are provided in order to get a survey of the vector lattices with different properties and to have an easy and quick access. Some examples of Sections 3.3, 6.2 and of Chapter 4 are omitted in this list. The abbreviations are clear, e. g., Φ𝑖 stands for Φ𝑖 (𝐸), 𝑖 = 1, 2, 3 and (Σ) for 𝐸 being a vector lattice of type (Σ). (Φ), (𝛼), (𝜇) mean that the vector lattice 𝐸 of continuous functions satisfies these conditions, respectively. We write 𝑆 ≃ 𝑇 if the two topological spaces 𝑆 and 𝑇 are homeomorphic.

Vector lattice and its properties

Number of Example

Page

Kaplansky K, description of Φ1 and Φ2 , Φ1 ≠ Φ2 ≠ K

3.4

23

(Σ), {0} ≠ Φ1 ≠ 𝐸

3.5

25

Dedekind complete, 𝐴𝑀, without order unit, Φ1 = Φ2 = 𝐸, each principal band has an order unit

3.20

32

Norm closed vector sublattice 𝐻 of a Banach lattice 𝐸 with Φ1 (𝐻) ≠ {0} and Φ1 (𝐸) = {0}

3.23

34

Closed ideal 𝐻 in a Banach lattice 𝐸 with Φ1 (𝐸) ∩ 𝐻 ⊈ Φ1 (𝐻)

3.25

35

Band 𝐻 in 𝐸 = 𝐶[0, 2] with Φ1 (𝐸) ∩ 𝐻 ⊈ Φ1 (𝐻)

3.26

35

Φ𝑘 (s) = c00 for 𝑘 = 1, 2, 3

3.48

48

The selfmajorizing elements in c

6.28

129

The selfmajorizing elements in ℓ∞

3.47

47

The selfmajorizing elements in 𝐵[0, 1] (real bounded functions on [0, 1])

3.51

49

No nontrivial selfmajorizing elements in 𝐶(ℝ)

3.49

48

𝐶𝑏(ℝ), Archimedean 𝑓-algebra, with multiplicative unit, Φ1 ≠ {0}, no invertible elements

3.61

56

(𝑟)-complete 𝑓-algebra, weak factorization property, without factorization property

3.63

57

Semiprime 𝑓-algebra without weak factorization property

3.64

58

Archimedean semiprime (𝑟)-complete 𝑓-algebra with a finite element in the product, but not all factors are not finite

3.78

68

208 | List of Examples

Vector lattice and its properties

Number of Example

Page

(Σ), with nonregular space (M(𝐸), 𝜏ℎ𝑘 )

5.17

110

(Σ), with prime ideals which are not maximal

5.19

114

(𝑟)-complete, (Σ), Φ1 = Φ2 , Φ2 is (Σ), MΦ = M, but MΦ ≠ M, (i. e., MΦ ≠ MΦ )

6.34

138

(𝑟)-complete, (Σ), Φ2 is not (Σ), (i. e., MΦ not 𝜎-compact) MΦ ≠ MΦ

6.35

140

Not (𝑟)-complete, (Σ), Φ1 = Φ2 , Φ2 is (Σ), MΦ ≠ MΦ

6.37

141

(𝑟)-complete, (Σ), Φ1 ≠ Φ2 , Φ2 is (Σ), MΦ ≠ MΦ

6.39

142

A maximal ideal from Φ1 can not be extended to 𝐸

6.40

142

Dedekind complete, (Σ), without any monotone norm

7.22

152

Almost periodic functions 𝑎P(ℝ), (Σ), not containing finite functions, Φ1 strongly separates the points of ℝ

8.2

157

A finite function which is not a finite element

8.14

163

𝐸 ⊂ 𝐶(𝑇), (Σ), some finite elements are not finite functions (𝜇), but not (𝐶𝑀 )

8.15

163

𝐸 ⊂ 𝐶(𝑇), (Σ), 𝑇 is 𝜎-compact, (Φ), (𝛼), 𝑇 ≃ MΦ , not (𝑟)-complete, M ≠ MΦ , Φ𝛼- but not E-representation

9.5

172

Existence of a noncompletely regular (⋆) Φ 𝛼 𝑐0 E-representation of a not (𝑟)-complete vector lattice of type (Σ)

9.7

176

𝐸 ⊂ 𝐶(𝑇), not (𝑟)-complete, (Σ), (𝜇), 𝐸 not solid in 𝐶(𝑇)

9.21

186

𝐸 ⊂ 𝐶𝑀 ([1, ∞), (𝑡𝑛 )𝑛∈ℕ ), (⋆), (𝜇), 𝐸 is its own e-representation, but not E-representation

9.25

190

𝐸 with monotone norm, under which no discrete functional is continuous (i. e., no maximal ideal is closed)

9.26

190

A Φ-basis which is not an 𝑅-basis

10.7

194

(Σ), (𝜇), with (𝑟)-complete 𝜎-basis which is not normal

10.21

201

List of Symbols ℓ∞ ℓ𝑝 A L(𝑋, 𝑌) S(𝐸) Δ(𝐸) AM(𝐸, 𝐹) DP(𝐸, 𝐹) F(𝐸, 𝐹) K(𝐸, 𝐹) K(𝑄) W(𝐸, 𝐹) K M(𝐸) MΦ (𝐸) int (𝐴) 𝜅 c0 c00 c s F(𝐴) L (𝑋, 𝑌) L 𝑏 (𝑋, 𝑌) L 𝑛 (𝑋, 𝑌) L 𝑟 (𝑋) L 𝑟 (𝑋, 𝑌) L+ (𝑋, 𝑌) ℕ≥𝑚 𝜔 Φ1 (𝐸) Φ2 (𝐸) Φ3 (𝐸) Π𝑝 (A) ℝ>𝜆 ℝ≥𝜆 Σ𝑝 (A) 𝜏ℎ𝑘 𝑐 𝜋, 𝜋𝑐 𝑎P(ℝ) 𝐵 [0, 1]

vector lattice of all real bounded sequences vector lattice of all real 𝑝-summable sequences lattice ordered algebra, ℓ-algebra space of all linear continuous operators from 𝑋 into 𝑌 proper space of the vector lattice 𝐸 the set of all discrete functionals of the vector lattice 𝐸 set of all 𝐴𝑀-compact operators 𝐸 → 𝐹 set of all Dunford-Pettis operators 𝐸 → 𝐹 set of all finite rank operators from 𝐸 → 𝐹 space of all compact operators between the Banach lattices 𝐸 and 𝐹 vector lattice of all real continuous functions on 𝑄 with compact support set of all weakly compact operators 𝐸 → 𝐹 Kaplansky vector lattice set of all maximal ideals of the vector lattice 𝐸 largest locally compact subspace in M(𝐸) the set of all interior points of the set 𝐴 in a topological space standard map from 𝑄 to M(𝐸) vector lattice of all real sequences converging to zero vector lattice of all sequences with a finite number of nonzero components vector lattice of all real converging sequences vector lattice of all real sequences set of all finite subsets of 𝐴 set of all linear operators from 𝑋 into 𝑌 set of all order bounded operators from 𝑋 to 𝑌 set of all order continuous operators from 𝑋 to 𝑌 set of all regular operators on 𝑋 set of all regular operators from 𝑋 to 𝑌 set of all positive operators in L (𝑋, 𝑌) set of all natural numbers 𝑛 ≥ 𝑚 canonical map from 𝐺(𝐸0 ) to M(𝐸0 ), where 𝐸0 is an ideal in 𝐸 ideal of all finite elements of the vector lattice 𝐸 ideal of all totally finite elements of the vector lattice 𝐸 ideal of all selfmajorizing elements of the vector lattice 𝐸 the set of all 𝑝-fold products in A for 𝑝 ∈ ℕ≥2 set of all real numbers 𝛼 > 𝜆 set of all real numbers 𝛼 ≥ 𝜆 set of all 𝑝-fold powers of positive elements of A for 𝑝 ∈ ℕ≥2 hull-kernel topology left, right multiplication by the element 𝑐 almost periodic functions vector lattice of all real bounded functions on [0, 1]

5 5 50 11 102 104 87 87 77 84 18 87 23 104 122 12 146 5 20 5 48 39 5 6 6 6 6 6 4 116 19 22 44 63 4 4 63 107 62 158 49

210 | List of Symbols 𝐶(𝑄) 𝐶∞ (𝑄) 𝐶0 (𝑄) 𝐶𝑏 (ℝ) 𝐶𝑀 (𝑄, (𝑒𝜈 )𝜈∈ℕ ) 𝐸̃ 𝐼𝐴 𝐼𝑥 𝐿 ∞ (Ω, Σ, 𝜇) 𝐿 𝑝 (Ω, Σ, 𝜇) 𝑆(𝐸) Orth(𝐸) suppM (𝑥) (M(𝐸), 𝜏ℎ𝑘 )

vector lattice of all real-valued continuous functions on the toplogical space 𝑄 vector lattice of all extended real-valued continuous functions on 𝑄 vector lattice of all real-valued continuous functions on the topological space 𝑄 vanishing at infinity vector lattice of all bounded real-valued continuous functions on ℝ vector lattice of slowly growing functions on 𝑄 order dual of the vector lattice 𝐸 ideal generated by the subset 𝐴 principal ideal (generated in a vector lattice by the element 𝑥) ordered vector space of all essentially 𝜇-bounded functions on Ω vector lattice of all real 𝜇-measurable functions on Ω such that ∫Ω |𝑥(𝑡)|𝑝 d𝜇 < ∞ set of all selfmajorizing elements in the vector lattice 𝐸 set of all orthomorphisms on the vector lattice 𝐸 abstract support of the element 𝑥 topological space of all maximal ideals of the vector lattice 𝐸

5 101 17 56 164 10 8 8 5 5 42 15 107 107

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Index (⋆)-representation, 145 (𝑜)-convergence, 7 (𝑟)-complete, 8 (𝑟)-convergence, 7 (𝑟)-dense, 187 (𝑟)-separable, 204 𝐴𝐿-space, 14 𝐴𝑀-space, 14 𝐶0(𝑄), 17 𝐾𝐵-space, 82 𝐿𝐹-space, 183 𝐿𝐹-vector lattice, 183 𝑅-basis, 195 𝑊-basis, 203 Φ-basis, 192 Φ-representation, 171 Φ𝛼-representation, 171 𝛼-representation, 171 ℓ-algebra, 50 𝛾-set, 173 𝛾-ultrafilter, 173 𝜑-set, 139, 173 𝜑-ultrafilter, 173 𝜓-set, 173 𝜓-ultrafilter, 173 𝜎-basis, 192 𝜎-representation, 148 𝑐0 -representation, 145 𝑑-algebra, 50 𝑓-algebra, 50 𝑙-ideal, 8 𝑝-th root of an element, 58 𝑢-norm, 14 abstract support, 107 algebra – lattice ordered, 50 – Riesz, 50 – semiprime, 50 – square-root closed, 57 – product of order 𝑝, 63 almost periodic function, 157 atom, 9 atomic vector lattice, 9 b-representation, 148 Banach lattice, 13 – direct sum, 39

band, 8 – generated by a set, 8 – principal, 8 – projection, 8 band projection, 8 basis – Φ-, 192 – 𝜎-, 192 – 𝑅-, 192 – 𝑊-, 203 – normal, 199 bipositive operator, 6 completion – Maeda-Ogasawara-Vulikh, 102 – universal, 102 component, 102 condition (𝜇), 184 condition (Φ), 163 condition (Σ󸀠 ), 9 condition (⋆), 105, 144 condition (𝑐0 ), 144 condition (𝛼), 157 condition (𝐶𝑀 ), 164 condition (A), 13 condition (B), 75 condition (C), 96 conditional representation, 119 cone, 4 – generating, 4 – reproducing, 4 convergence – order, 7 – uniform, 7 – with regulator, 7 Dedekind completion, 7 discrete functional, 104 disjoint, 8 E-representation, 147 e-representation, 147 element – decomposable, 202 – discrete, 9 – finite, 19 – invertible, 50

218 | Index – nilpotent, 50 – selfmajorizing, 41 – totally finite, 22 factorization property, 57 – weak, 57 Fatou norm, 96 filter, 115 Fréchet-lattice, 183 function – almost periodic, 157 – extended real-valued, 101 functional – discrete, 104 functions – slowly growing, 164 fundament, 101 ideal, 8 – maximal, 103 – of all selfmajorizing elements, 44 – of all finite elements, 19 – of all totally finite elements, 22 – order dense, 101 – prime, 106 – ring, 51 infimum, 5 lattice homomorphism, 10 lattice isomorphism, 10 lattice norm, 13 lattice operations, 13 Levi norm, 75 lower bound, 4 majorant, 19 majorizing sublattice, 33 map – canonical, 116 – standard, 146 monotone norm, 13 multiplication – left, right, 62 Nakano norm, 17 Nakano space, 102 net – decreasing, 5 – increasing, 5

norm – Fatou, 96 – lattice, 13 – Levi, 75 – monotone, 13 – monotonically complete, 75 – Nakano, 17 – operator, 69 – order continuous, 13 – order unit, 14 – regular, 69 – Riesz, 13 – submultiplicative Riesz, 54 norm dual, 11 normal basis, 199 normed Riesz space, 13 normed vector lattice, 13 operator – 𝐴𝑀-compact, 87 – band preserving, 15 – bipositive, 6 – compact, 84 – Dunford-Pettis, 87 – interval preserving, 29 – Maharam, 29 – order bounded, 6 – order continuous, 6 – positive, 6 – regular, 6 – weakly compact, 87 operator norm, 69 order – coordinatewise, 5 – pointwise, 5 order unit, 9 order bounded set, 4 order continuous norm, 13 order dual, 10 order homomorphism, 6 order interval, 4 order isomorphism, 6 order unit – generalized, 26 order unit norm, 14 orthomorphism, 15 positive operator, 6 prime ideal, 106

Index

principal projection property, 8 product of order 𝑝, 63 property – (Σ󸀠 ), 9 – (𝐶𝑀 ), 164 – (𝑝𝑝𝑝), 8 – 𝜎-closure, 128 – (∗), 83 – (W1), 82 – factorization, 57 – Riesz decomposition, 5 – weak factorization, 57 R-representation, 195 radical-free, 105 regular operator, 6 regulator of convergence, 7 representation, 100, 144 – (⋆)-, 145 – 𝛼-, 171 – Φ-, 171 – Φ𝛼-, 171 – 𝜎-, 148 – 𝑐0 -, 145 – b-, 148 – by an 𝑅-basis, 195 – completely regular, 145 – conditional, 119 – E-, 147 – e-, 147 – R-, 195 representation basis, 192 Riesz decomposition property, 5 Riesz homomorphism, 10 Riesz isomorphism, 10 Riesz norm, 13 Riesz seminorm, 21 Riesz space, 6 Riesz-Kantorovich formulas, 10 ring ideal, 51 row, 139 semi-order unit, 41 semi-simple, 105 sequence – uniformly Cauchy, 8 set – 𝐺(𝐴), 107 – complete, 8

|

– majorized, 4 – normal, 8 – order bounded, 4 – regularly open, 35 – separating, 144 – solid, 8 – strongly separating, 144 – sufficient, 106 – total, 22 – total (of functionals), 105 – minorized, 4 space – 𝜎-compact, 123 – 𝐿𝐹-, 183 – compact extremally disconnected, 101 – of all maximal ideals, 107 – ordered vector, 4 – proper, 102 Stone space for 𝐸, 102 Stonian space, 101 strict inductive limit, 182 strong point, 103 subalgebra, 51 subspace MΦ (𝐸), 122 summable family, 39 supremum, 5 Theorem – Kakutani-Bohnenblust-Krein-Krein, 16 – Kakutani-Bohnenblust-Nakano, 17 – Kawai, 184 – Maeda-Ogasawara-Vulikh, 102 – Nakano, 17 – Nakano-Makarov, 15 – Riesz-Kantorovich, 9 topology – 𝜎-closure property, 128 – Lebesgue, 13 – hull-kernel, 107 ultrafilter, 115 unit – multiplicative, 50 – order, 9 – semi-order, 41 – strong local, 103 – weak order, 9 upper bound, 4

219

220 | Index vector lattice, 6 – 𝜎-Dedekind complete, 7 – 𝐿𝐹-, 183 – Archimedean, 6 – atomic, 9 – Dedekind complete, 7 – Kaplansky, 23 – normed, 13 – of bounded elements, 9

– of majorizing functions, 164 – of slowly growing functions, 164 – of type (Σ), 9 – of type (𝐶𝑀 ), 184 – radical-free, 105 – uniformly complete, 8 weakly sequentially complete, 82 weakly sequentially continuous, 13 wedge, 4