Banach Algebras and Applications: Proceedings of the International Conference held at the University of Oulu, July 3-11, 2017 (De Gruyter Proceedings in Mathematics) 311060132X, 9783110601329

Table of contents :
Dedication
Preface
Contents
Beurling’s Theorem on locally compact abelian groups
Fréchet algebras with a dominating Hilbert algebra norm
Relations between ideals of the Figà-Talamanca Herz algebra Ap(G) of a locally compact group G and ideals of Ap(H) of a closed subgroup
The composition of conditional expectation and multiplication operators
On the generation of Arveson weakly continuous semigroups
ℓ1-bases in Banach algebras and Arens irregularities in harmonic analysis
On spectral synthesis in convolution Sobolev algebras on the real line
Projective and free matricially normed spaces
Banach spaces whose algebras of operators are Dedekind-finite but they do not have stable rank one
Invariant Complementation Property and Fixed Point Sets On Power Bounded Elements in the Group von Neumann Algebra
Subspaces that can and cannot be the kernel of a bounded operator on a Banach space
Towards a sheaf cohomology theory for C*-algebras
Tame functionals on Banach algebras
Left Ideals of Banach Algebras and Dual Banach Algebras

Citation preview

Mahmoud Filali (Ed.) Banach Algebras and Applications

De Gruyter Proceedings in Mathematics

Banach Algebras and Applications Proceedings of the International Conference held at the University of Oulu, July 3—11, 2017 Edited by Mahmoud Filali

Editor Dr. Mahmoud Filali University of Oulu Dept. of Mathematical Sciences P.O.Box 8000 FIN-90014 Oulu Finland

ISBN 978-3-11-060132-9 e-ISBN (PDF) 978-3-11-060241-8 e-ISBN (EPUB) 978-3-11-060043-8

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Printing and binding:CPI books GmbH, Leck www.degruyter.com

| Dedication In the last two years the Banach Algebra community has sadly lost two distinguished members, Philip Curtis and Eberhard Kaniuth. Both, Philip and Eberhard had a very strong and positive impact on the mathematics community. Phil Curtis together with Bill Bade organized the first conference in this series in 1974 at UCLA. Eberhard Kaniuth has been one of the main pillars in Banach algebra theory and abstract harmonic analysis, and had a deep and wide impact on the work taking place in these two fields. These proceedings are dedicated to their memories.

VI |

Preface The 23rd conference on Banach Algebras and Applications was held in Oulu from July 3rd to July 11th, 2017. It is the most important scientific event in the Banach algebras community. The first conference in this sequence took place at the University of California, Los Angeles (UCLA) in 1974. This was followed by Leeds 1976, Long Beach 1981, Copenhagen 1985, Berkeley 1986, Leeds 1987, Berkeley 1988, Canberra 1989, Berkeley 1990, Cambridge 1991, Winnipeg 1993, Newcastle 1995, Blaubeuren 1997, Pomona 1999, Odense 2001, Edmonton 2003, Bordeaux 2005, Quebec 2007, Poznan 2009, Waterloo 2011, Gothenburg 2013, Fields Institute (Toronto) 2015. With more than 70 participants from America, Asia, Europe and Africa, it was probably one of the biggest and most diverse events in Mathematics in Northern Finland. Banach algebras is a multilayered area in mathematics and has many ramifications. The diversity of the schools taking part in the conference made the event very successful and exciting. The talks reflected recent achievements in many areas contained in Banach algebra theory such as Banach Algebras over Groups, Abstract Harmonic Analysis, Group Actions, Amenability, Topological Homology, Semigroup Compactifications, Arens Irregularity, C*-Algebras and Dynamical Systems, Operator Theory, Operator Spaces, and more. In fact the last decade has seen so much progress in many branches of Banach Algebra theory, and so much of these new achievements has been presented at the conference. We would like to thank the participants who came all the way to Oulu to attend the conference and share all the exciting mathematics they have been working on in recent years. The present volume contains sixteen refereed articles based on the high level expository talks presented at the conference. So, our thanks are due to the authors, there would be no proceedings without their contributions. We are also very thankful for the efforts and time spent by the referees checking thoroughly the papers, and for detailed reports and corrections sent to the authors. We believe the proceedings will ultimately serve as a platform for researchers in Banach algebras theory and related areas. Partial support was made available to us by The Mathematics Foundation (The Finnish Academy of Science and Letters). This has helped to support a number of colleagues who did not have any financial support from their home institute or country. We are gratefully indebted to the Foundation and particularly to Olli Martio for his quick and positive answer. Kiitos Olli! BusinessOulu with Riina Aikio and Helena Pikkarainen has played an important role for the success of the event and the comfort of our guests during the conference. These two wonderful ladies were behind the wonderful social program and

|

VII

Fig. 1: Created by Peetu Karttunen

many other practical matters. Their help and enthusiasm with the conference are gratefully acknowledged. The painstaking task of putting all the tex files sent by the authors into the form required by the journal was noticeably lifted off my shoulders by Nadja Schedensack, Project Editor of de Gruyter journal, and by our department technician Pekka Kangas. To both I wish to express my sincere gratitude for their continuous help and patience. Finally thanks to my family and friends who had been helping me right from the beginning long before the conference too place. Without their encouragements and their help with the practical matters, this conference would not have taken place. Mahmoud Filali Oulu, September 30, 2019

Contents Ali Baklouti and Mahmoud Filali Beurling’s Theorem on locally compact abelian groups | 1 Tomasz Ciaś Fréchet algebras with a dominating Hilbert algebra norm | 5 Antoine Derighetti Relations between ideals of the Figà-Talamanca Herz algebra A p (G) of a locally compact group G and ideals of A p (H) of a closed subgroup | 35 Y. Estaremi The composition of conditional expectation and multiplication operators | 47 Jean Esterle On the generation of Arveson weakly continuous semigroups | 57 Mahmoud Filali and Jorge Galindo ℓ1 -bases in Banach algebras and Arens irregularities in harmonic analysis | 95 José E. Galé, María M. Martínez and Pedro J. Miana On spectral synthesis in convolution Sobolev algebras on the real line | 133 A. Ya. Helemskii Projective and free matricially normed spaces | 151 Bence Horvàth Banach spaces whose algebras of operators are Dedekind-finite but they do not have stable rank one | 165 Anthony To-Ming Lau Invariant Complementation Property and Fixed Point Sets On Power Bounded Elements in the Group von Neumann Algebra | 177

X | Contents Niels Jakob Laustsen and Jared T. White Subspaces that can and cannot be the kernel of a bounded operator on a Banach space | 189 Martin Mathieu Towards a sheaf cohomology theory for C*-algebras | 197 Michael Megrelishvili Tame functionals on Banach algebras | 213 Jared T. White Left Ideals of Banach Algebras and Dual Banach Algebras | 227

Ali Baklouti and Mahmoud Filali

Beurling’s Theorem on locally compact abelian groups Abstract: We prove the analogue to Beurling’s theorem for any locally compact abelian group. This generalizes an earlier work by the first author and Kaniuth on Hardy’s uncertainty principle (cf. [1]). Keywords: Beurling’s theorem, locally compact abelian group. Classification: Primary 22A05; Secondary 54E15 54H11.

1 Introduction An attractive theorem of Beurling on Fourier transform pairs says that if f ∈ L1 (R) and (1) ∫ ∫ |f(x)||̂f (y)|exp(|xy|)dxdy < ∞, R R

then f = 0. In other words, the trivial function is the only function in L1 (R) for which f ̂f is integrable on R2 with respect to the measure exp(|xy|)dxdy. Here, ̂f is the Fourier transform of f. The theorem is stated without a proof on page 372 of Beurling’s collected works [2]. Based on the notes Hörmander took when Beurling explained the result to him in the mid-sixties, Hörmander reproduced the proof and published it in [5]. This was followed by a long list of papers extending Beurling’s theorem to various groups. This list is too long to cite all the papers, but the reader may consult [3], where Beurling’s Theorem was generalized to R n , and which is needed for our proofs. In this note, we give an analogue to Beurling’s assumption (1) and prove the analogue to Beurling’s theorem for any locally compact abelian group G.

The second author wishes to express his sincere gratitude to Sfax University for the partial financial support. Ali Baklouti, Département de Mathématiques, Université de Sfax, Sfax, Tunisie, mail: [email protected] Mahmoud Filali, Department of Mathematical Sciences, University of Oulu, 90014 Oulu, Finland, email: mahmoud.filali@oulu.fi https://doi.org/10.1515/9783110602418-0001

2 | Baklouti-Filali

2 The theorem Note first that Beurling’s theorem does not hold if G is discrete. For instance, take the function f = δ e and let Ψ be any function on G × ̂ G such that Ψ(e, .) is measurable and bounded on ̂ G. Then ̂f is the constant function 1 on ̂ G and ∑ ∫ f(x)̂f (y)Ψ(x, y)dxdy = ∫ Ψ(e, y)dy < ∞.

x∈G ̂ G

̂ G

So we are actually concerned with non-discrete groups. We denote the scalar product of x and y in R n simply by xy. We use the structure theorem and write our group as G = R n × H, where n ≥ 0 and H is a locally compact abelian group which contains an open compact subgroup K, see for example [4, Theorems 24.29, 24.30]. Write u in G as u = (x, s) with x ∈ R n and s ∈ H. Write χ in ̂ G as χ = (y, ξ) with n ̂ y ∈ R and ξ ∈ H. and let φ : G → R n and ψ : ̂ G → R n be given, respectively, by {x, φ(u) = { 0, {

if if

n>0

n=0

and

{y, ψ(χ) = { 0, {

if if

n>0

n = 0.

For the analogue to Beurling’s assumption (1), we set B(f) := ∫ ∫ |f(u)||̂f (χ)|exp(|φ(u)φ(χ)|)dudχ.

(2)

G ̂ G

Then, the analogue to Beurling’s theorem may be stated as follows: Theorem 2.1. Let G be a locally compact abelian group with dual group ̂ G and let f ∈ L1 (G). Then the implication B(f) < ∞ 󳨐⇒ f = 0 holds if and only if the connected component of the identity in G is not compact. Proof. Suppose that the component of the identity e in G is not compact, i.e., n > 0, and write as above our group as G = R n × H. Note first that the assumption B(f) < ∞ means that ∫ ∫ ∫ ∫ |f(x, s)||̂f (y, ξ)|exp(|xy|)dxdydsdξ < ∞,

Rn Rn H H ̂

and so ∫ ∫ ∫ |f(x, s)||̂f (y, ξ)|exp(|xy|)dxdyds < ∞ Rn

Rn

H

(3)

Beurling’s Theorem on locally compact abelian groups |

3

̂ Consider now for each x ∈ R n , the function almost everywhere on H. F(x, ξ) = ∫ f(x, s)ξ(s)ds, H

i.e., F(x, .) is the Fourier transform of the function f x defined on H by f x (s) = f(x, s), and note that (4) |F(x, ξ)| ≤ ∫ |f(x, s)ξ(s)|ds = ∫ |f(x, s)|ds. H

H

̂ the function F ξ : R n → R, given by F ξ (x) = F(x, ξ), is in Then for each ξ ∈ H, 1 n L (R ) since ∫ |F ξ (x)|dx ≤ ∫ ∫ |f(x, s)ξ(s)|dsdx = ∫ |f(u)|du, Rn

Rn H

G

and its Fourier transform is given, for every y ∈ R n , by F̂ξ (y) = ∫ F ξ (x)exp(ixy)dx = ∫ (∫ f(x, s)ξ(s)ds) exp(ixy)dx = ∫ ∫ f(x, s)ξ(s)exp(ixy)dxds = ̂f (y, ξ).

Rn

Rn

H

(5)

Rn H

The observations (3), (4) and (5) lead to ∫ ∫ |F ξ (x)||F̂ξ (y)|exp(|xy|)dxdy

Rn Rn

≤ ∫ ∫ ∫|f(x, s)|̂f (y, ξ)exp(|xy|)dxdyds < ∞ Rn Rn H

̂ almost everywhere on H. Accordingly, we may apply Beurling’s Theorem when n = 1 or [3] when n > 1 ̂ F ξ (x) = ̂f x (ξ) = 0 almost everywhere on to deduce, that for almost every ξ ∈ H, n R . Therefore, f x = 0 for almost every x in R n . This implies that f(x, s) = 0 almost everywhere on G = R n × H, as required. Conversely, suppose that the component of e in G is compact. Then, G = Z m × M, where M contains an open compact subgroup K. Note that here φ(u) = ψ(χ) = 0 for every u ∈ G and χ ∈ ̂ G since n = 0. We follow the notation in [4, Sections 23-24]. Let A(̂ G, K) be the annihilator of ̂ K in G, that is, A(̂ G, K) = {χ ∈ ̂ G : χ|K = 1}.

4 | Baklouti-Filali By [4, Remarks 23.24 or 23.29], A(̂ G, K) is a compact subgroup of ̂ G since K is open. (The converse is also true as noted in 23.29). Let the Haar measure on G be normalized on K, and let f be the function with values 1 on K and 0 elsewhere. Then f is a non-trivial function in L1 (G). Moreover, for every χ ∈ ̂ G,

Accordingly,

{ ̂f (χ) = ∫ f(u)χ(u)du = ∫ χ(u)du = 0, if χ|K ≠ 1 { 1 otherwise. K G {

∫ ∫ |f(u)||̂f (χ)|exp(|φ(u)ψ(χ)|)dudχ = ∫ ∫ ̂f (χ)dudχ = (∫ du) (∫ ̂f (χ)dχ) G ̂ G

K ̂ G

K

= (∫ du) ( ∫ K

since A(̂ G, K) is compact. This completes the proof.

A(̂ G,K)

̂ G

dχ) < ∞

Bibliography [1]

A. Baklouti and E. Kaniuth, On Hardy’s uncertainty principle for solvable locally compact groups. J. Fourier Anal. Appl. 16, No. 1, (2010), 129-147.

[2]

A. Beurling, The collected works of Arne Beurling, Vol. 1–2, Complex analysis. Edited by L. Carleson, P. Malliavin, J. Neuberger and J. Wermer. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, (1989).

[3]

A. Bonammi, B. Demange and P. Jaming, Hermite functions and uncertainty principles for the Fourier and windowed Fourier transforms, Rev. Mat. Iberoamericana 19 (2003), 23–55.

[4]

E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups, integration theory, group representations, Academic Press Inc., Publishers, New York (1963).

[5]

L. Hörmander, A uniqueness theorem of Beurling for Fourier transform pairs, Ark. Mat. 29 (1991), 237–240.

Tomasz Ciaś

Fréchet algebras with a dominating Hilbert algebra norm Abstract: Let L∗ (s) be the maximal O∗ -algebra of unbounded operators on ℓ2 whose domain is the space s of rapidly decreasing sequences. This is a noncommutative topological algebra with involution which can be identified, for instance, with the algebra L(s) ∩ L(s󸀠 ) or the algebra of multipliers for the algebra L(s󸀠 , s) of smooth compact operators. We give a simple characterization of unital commutative Fréchet ∗ -subalgebras of L∗ (s) isomorphic as Fréchet spaces to nuclear power series spaces Λ∞ (α) of infinite type. It appears that many natural Fréchet ∗ -algebras are closed ∗ -subalgebras of L∗ (s), for example, the algebras C ∞ (M) of smooth functions on smooth compact manifolds and the algebra S(R n ) of smooth rapidly decreasing functions on R n . Keywords: Representations of commutative Fréchet algebras with involution, topological algebras of unbounded operators, nuclear Fréchet algebras of smooth functions, dominating norm, Hilbert algebra. Classification: Primary: 46J25. Secondary: 46A11, 46A63, 46E25, 46K15, 47L60.

1 Introduction Let s be the Fréchet space of rapidly decreasing complex sequences and let L∗ (s) := {x : s → s : x is linear, s ⊂ D(x∗ ) and x∗ (s) ⊂ s}, where D(x∗ ) is the domain of the adjoint of an unbounded operator x on ℓ2 . The class L∗ (s) is known as the maximal O∗ -algebra with domain s and it can be seen as the largest ∗ -algebra of unbounded operators on ℓ2 with domain s – for details see the book of Schmüdgen [18, Section I.2.1]. The ∗ -algebra L∗ (s) can be topologised in several natural ways, as is shown in [18, Sections I.3.3 and I.3.5]. Here

The research of the author was supported by the National Center of Science (Poland), grant no. 2013/10/A/ST1/00091. Tomasz Ciaś , Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań, Umultowska 87, 61-614 Poznań, Poland, e-mail: [email protected] https://doi.org/10.1515/9783110602418-0002

6 | Ciaś the space L∗ (s) is considered with – the best from the functional analysis point of view – locally convex topology τ∗ (for definition see Preliminaries and also Proposition 2.6). Indeed, standard tools of functional analysis, such as closed graph theorem, open mapping theorem or uniform boundedness principle, can be applied to (L∗ (s), τ∗ ) (see [8, Th. 4.5]). Furthermore, L∗ (s) is a topological ∗ -algebra – i.e. multiplication is separately continuous and involution is continous – but it is neither locally m-convex nor a Q-algebra. The algebra L∗ (s) is isomorphic as a topological ∗ -algebra, for example, to the algebra L(s) ∩ L(s󸀠 ), the algebra of multipliers for the algebra L(s󸀠 , s) of smooth compact operators and also to the matrix algebra Λ(A) := {x = (x ij ) ∈ CN : ∀N ∈ N ∃n ∈ N 2

∑ |x ij | max {

i,j∈N2

iN jN , } < ∞}; jn in

for details and more information about topological and algebraic properties of L∗ (s) we refer the reader to [8]. The space s carries all the information about nuclear Fréchet (even locally ¯ ¯ convex) spaces. Indeed, by the Komura-K omura theorem, a Fréchet space is nuclear if and only if it is isomorphic to some closed subspace of sN (see [14, Cor. 29.9]). What about closed subspaces of the space s itself? In [21] Vogt proved that a nuclear Fréchet space is isomorphic to a closed subspace of s if and only if it has the so-called property (DN). Moreover, quotients of s were characterised by Vogt and Wagner in [22] via the so-called property (Ω). Consequently, we have the following characterization: a nuclear Fréchet space is isomorphic to a complemented subspace of s if and only if it has the properties (DN) and (Ω). It is also well-known that a Fréchet space with (DN), (Ω) and a Schauder basis is isomorphic to a power series space Λ∞ (α) of infinite type. However, it is still an open problem – a particular case of the famous Mityagin-Pełczyński problem – whether there is a complemented subspace of s without a basis. In this paper, we are mainly interested in unital Fréchet algebras with involution which are isomorphic as Fréchet spaces to nuclear power series spaces of infinite type. We show that a large class of them – those algebras E which admit a dominating Hilbert norm || ⋅ || = √(⋅, ⋅) such that (xy, z) = (y, x∗ z)

(1)

for all x, y, z ∈ E – can be embedded into L∗ (s) as closed, even complemented, ∗ subalgebras (see Theorem 3.5 and Remark 3.17). In the commutative case we even have the following characterization: a unital commutative Fréchet ∗ -algebra isomorphic as a Fréchet space to a nuclear power series space Λ∞ (α) of infinite type is isomorphic as a Fréchet ∗ -algebra to a closed ∗ -subalgebra of L∗ (s) if and only if it

Fréchet algebras | 7

admits a dominating Hilbert norm satisfying condition (1) (see again Theorem 3.5). In Theorem 3.6 we also characterize commutative Fréchet unital ∗ -subalgebras of L∗ (s) consisting of bounded operators on ℓ2 and isomorphic as Fréchet space to nuclear spaces Λ∞ (α). It is worth noting that condition (1) appears in the definiton of Hilbert algebras playing an important role in the theory of von Neumann algebras (see [9, A.54]). The above-mentioned results may be seen as a step towards an analogue – in the context of nuclear power series spaces of infinite type – of the celebrated commutative Gelfand-Naimark theorem. In the separable case it states that there is one to one correspondence (given by isometric ∗ -isomorphisms) between Banach algebras C(K) of continuous functions on compact Hausdorff metrizable spaces K and closed unital commutative ∗ -subalgebras of the C∗ -algebra B(ℓ2 ) of bounded operators on ℓ2 . Our results are applicable. In the last section we give concrete examples of Fréchet ∗ -algebras which can be represented in L∗ (s) in the way described above. Among them there are: the algebras C∞ (M) of smooth functions on smooth compact manifolds, the algebras E(K) with Schauder basis of smooth Whitney jets on compact sets K with the extension property, the algebra S(R n ) of smooth rapidly decreasing functions on R n , nuclear power series algebras Λ∞ (α) of infinite type and the noncommutative algebra L(s󸀠 , s) of compact smooth operators. We also provide one counterexample. We show that the unital commutative Fréchet ∗ -algebra A ∞ (D) of holomorphic functions on the open unit disc D with smooth boundary values is not isomorphic to any closed ∗ -subalgebra of L∗ (s).

2 Preliminaries The canonical ℓ2 norm and the corresponding scalar product will be denoted by || ⋅ ||ℓ2 and ⟨⋅, ⋅⟩, respectively. For locally convex spaces E and F, we denote by L(E, F) the space of all continuous linear operators from E to F and we set L(E) := L(E, E). These spaces will be considered with the topology τL(E,F) of uniform convergence on bounded sets. By a topological ∗ -algebra E we mean a topological vector space endowed with at least separately continuous multiplication and continuous involution which make E a ∗ -algebra. A Fréchet ∗ -algebra is a topological ∗ -algebra whose underlying topological vector space is a Fréchet space (i.e. metrizable complete locally convex space). We do not require a Fréchet ∗ -algebra to be locally m-convex.

8 | Ciaś Let α = (α j )j∈N be a monotonically increasing sequence in (0, ∞) such that limj→∞ α j = ∞. Then Λ∞ (α) := {(ξ j )j∈N ⊂ CN : |ξ|2α,q := ∑ |ξ j |2 e2qα j < ∞ ∞

j=1

for all q ∈ N0 }

equipped with the norms | ⋅ |α,q , q ∈ N0 , is a Fréchet space and it is called a power series space of infinite type. It appears that the space Λ∞ (α) is nuclear if and only if j supj∈N log α j < ∞ (see e.g. [14, Prop. 29.6]). In particular, for the sequence α j := log j, j ∈ N, we obtain the space s of rapidly decreasing sequences, i.e. s := {(ξ j )j∈N ∈ CN : |ξ|2q := ∑ |ξ j |2 j2q < ∞ j∈N

for all q ∈ N0 }.

(2)

By s n we denote the Hilbert space corresponding to the norm | ⋅ |n . The strong dual of s – i.e. the space of all continuous linear functionals on s with the topology of uniform convergence on bounded subsets of s (see e.g. [14, Def. on p. 267]) – is isomorphic to the space s󸀠 := {(ξ j )j∈N ∈ CN : |ξ|2−q := ∑ |ξ j |2 j−2q < ∞ j∈N

for some q ∈ N0 }

(3)

of slowly increasing sequences equipped with the inductive limit topology for the sequence (s−n )n∈N0 of Hilbert spaces corresponding to the norms | ⋅ |−n . In other words, the locally convex topology on s󸀠 is given by the family {| ⋅ |󸀠B }B∈B of seminorms , |ξ|󸀠B := supη∈B |⟨η, ξ⟩|, where B denotes the class of all bounded subsets of s and, recall, ⟨⋅, ⋅⟩ is the canonical scalar product on ℓ2 . Definition 2.1. A Fréchet space E with a fundamental system (|| ⋅ ||q )q∈N0 ) of seminorms (1) has the property (DN) (cf. [14, Def. on p. 359]) if there is a continuous norm ||⋅|| on E – called a dominating norm – such that for all q ∈ N0 there is r ∈ N0 and C > 0 such that ||x||2q ≤ C||x|| ||x||r for all x ∈ E; (2) has the property (Ω) (cf. [14, Def. on p. 367]) if for all p ∈ N0 there is q ∈ N0 such that for all r ∈ N there are θ ∈ (0, 1) and C > 0 with ||y||∗q ≤ C||y||∗p 1−θ ||y||∗r θ for all y ∈ E󸀠 , where E󸀠 is the topological dual of E and ||y||∗p := sup{|y(x)| : ||x||p ≤ 1}.

Fréchet algebras | 9

The properties (DN) and (Ω) are linear-topological invariants which play a key role in a structure theory of nuclear Fréchet spaces. The following Theorem is due to Vogt and Wagner. Theorem 2.2. ([14, Ch. 31] and [21, 22]) A Fréchet space is isomorphic to: (i) a closed subspace of s if and only if it is nuclear and has the property (DN); (ii) a quotient of s if and only if it is nuclear and has the property (Ω); (iii) a complemented subspace of s if and only if it is nuclear and has the properties (DN) and (Ω). We also cite another result of Vogt which will be crucial for our futher considerations. Theorem 2.3. [23, Cor. 7.7] Let E be a Fréchet space isomorphic to a power series space Λ∞ (α) of infinite type. Then for every dominating Hilbert norm || ⋅ || on E there is an isomorphism u : E → Λ∞ (α) such that ||uξ||ℓ2 = ||ξ|| for all ξ ∈ E. Let E be a Fréchet space with a continuous Hilbert norm || ⋅ ||. Let H be the completion of E in the norm || ⋅ || and let (⋅, ⋅) be the corresponding scalar product. Then we define L∗ (E, || ⋅ ||) := {x : E → E : x is linear, E ⊂ D(x∗ ) and x∗ (E) ⊂ E}, where

D(x∗ ) := {η ∈ H : ∃ζ ∈ H ∀ξ ∈ E

(xξ, η) = (ξ, ζ)}

and x∗ η := ζ for η ∈ D(x∗ ). In the case when E is a closed subspace of s or E = Λ∞ (α) we write L∗ (E) instead of L∗ (E, || ⋅ ||ℓ2 ). Since E is a dense linear subspace of H, each x ∈ L∗ (E, || ⋅ ||) can be considered as a dense unbounded operator in H with domain D(x) = E, and thus it has the adjoint x∗ : D(x∗ ) → H. By definition, the operator x∗ |E , for simplicity denoted again by x∗ , is in L∗ (E, || ⋅ ||), as well. Moreover, by definition, D(xy) := {ξ ∈ D(y) : yξ ∈ D(x)} = E for all x, y ∈ L∗ (E, || ⋅ ||). This shows that L∗ (E, || ⋅ ||) is a ∗ -algebra. In fact, the class L∗ (E, || ⋅ ||) can be seen as the largest ∗ -algebra of unbounded operators on H with domain E and it is known as the maximal O∗ -algebra with domain E (see [18, 2.1] for details). In the theory of maximal O∗ -algebras – and, more generally, of algebras of unbounded operators in Hilbert spaces – one consider the so-called graph topology ([18, Def. 2.1.1]). With E and || ⋅ || as above, the graph topology of L∗ (E, || ⋅ ||) on E is, by definition, given by the system of seminorms (|| ⋅ ||a )a∈L∗ (E,||⋅||) , ||ξ||a := ||aξ|| for ξ ∈ E.

10 | Ciaś The following easy observation is kind of folklor – for completness we present here the proof. Proposition 2.4. Let E be a Fréchet space with a continuous Hilbert norm ||⋅||. Then the graph topology of L∗ (E, || ⋅ ||) on E is weaker than the Fréchet space toplogy. Proof. Let (⋅, ⋅) denote the scalar product corresponding to the Hilbert norm || ⋅ || and let H be the completion of E in the norm || ⋅ ||. We shall show that each a ∈ L∗ (E, || ⋅ ||) is a continuous map from the Fréchet space E to the Hilbert space H. Let (ξ j )j∈N ⊂ E be a sequence converging in the Fréchet space topology to 0 and assume that aξ j converges in the norm || ⋅ || to some η ∈ H. We have, for all ζ ∈ E, lim (aξ j , ζ) = (η, ζ)

j→∞

and, on the other hand, lim (aξ j , ζ) = lim (ξ j , a∗ ζ) = 0.

j→∞

j→∞

Hence, (η, ζ) = 0 for all ζ ∈ E, and thus η = 0. Consequently, by the closed graph theorem for Fréchet spaces (cf. [14, Th. 24.31]), the map a : E → H is continuous, which is the desired conclusion. Sometimes the initial Fréchet space topology and the graph topology coincide. Proposition 2.5. Let E be a Fréchet space isomorphic to a power series space Λ∞ (α) of infinite type and let || ⋅ || be a dominating Hilbert norm on E. Then the graph topology of L∗ (E, || ⋅ ||) on E coincides with the Fréchet space topology. Proof. Let (⋅, ⋅) denote the scalar product corresponding to the Hilbert norm || ⋅ ||. By [23, Cor. 7.7], there is an isomorphism u : E → Λ∞ (α) such that ||uξ||ℓ2 = ||ξ|| for all ξ ∈ E. Let ||ξ||n := |uξ|α,n for ξ ∈ E and n ∈ N. Then (|| ⋅ ||n )n∈N is a fundamental sequence of dominating Hilbert norms on E. For n ∈ N, we define the diagonal map d n : Λ∞ (α) → Λ∞ (α), d n ξ := (e nα j ξ j )j∈N . Clearly, each d n is an automorphism of the Fréchet space Λ∞ (α) and ||d n ξ||ℓ2 = |ξ|α,n for all ξ ∈ Λ∞ (α). Now, for n ∈ N, let a n : E → E, a n := u−1 d n u. We have (a n ξ, ζ) = (u−1 d n uξ, ζ) = ⟨d n uξ, uζ⟩ = ⟨uξ, d n uζ⟩ = (ξ, u−1 d n uζ) = (ξ, a n ζ) for all ξ, ζ ∈ E, whence a n ∈ L∗ (E, || ⋅ ||). Consequently, since ||ξ||n = ||a n ξ|| for all ξ ∈ E, i.e. || ⋅ ||n = || ⋅ ||a n , the graph topology of L∗ (E, || ⋅ ||) on E is finer than the Fréchet space toppology and thus, in view of Proposition 2.4, these topologies are equal. There are plenty natural topologies on the space L∗ (E, ||⋅||) (see [18, Sect. 3.3, 3.5]). Here we are interested in the locally convex topology τ∗ on L∗ (E, || ⋅ ||) given by

Fréchet algebras |

the seminorms

11

p a,B (x) := max { sup ||axξ||, sup ||ax∗ ξ||}, ξ ∈B

ξ ∈B

where a and B run over L∗ (E, || ⋅ ||) and the class of all bounded subsets of E equipped with the graph topology of L∗ (E, || ⋅ ||), respectively (see [18, pp. 81–82]). It is well-known that L∗ (E, || ⋅ ||) endowed with the topology τ∗ is a topological ∗ -algebra (cf. [18, Prop. 3.3.15 (i)]). If we, moreover, assume that E is isomorphic to a power series space of infinite type and || ⋅ || is a dominating Hilbert norm on E, then, by Proposition 2.5 and [18, Prop. 3.3.15 (iv)], (L∗ (E, || ⋅ ||), τ∗ ) is complete. The following characterization of the topology τ∗ is a direct consequence of Proposition 2.5. Proposition 2.6. Let E be a Fréchet space isomorphic to a power series space of infinite type and let || ⋅ || be a dominating Hilbert norm on E. Let (|| ⋅ ||n )n∈N be a fundamental sequence of norms on E. Then the topology τ∗ on L∗ (E, || ⋅ ||) is given by the seminorms (p n,B )n∈N,B∈BE , p n,B (x) := max{sup ||xξ||n , sup ||x∗ ξ||n }, ξ ∈B

ξ ∈B

(4)

where BE denote the class of all bounded subsets of E.

3 Fréchet subalgebras of L∗ (s) In this section we give abstract descriptions of two large classes of complemented commutative Fréchet ∗ -subalgebras of L∗ (s) (Theorems 3.5 and 3.6). Moreover, we provide a criterion for the existence of a “nice” embedding in L∗ (s) of not necessarily commutative Fréchet ∗ -algebras (see Remark 3.17). Let us first recall the notion of Hilbert algebras. Definition 3.1. (cf. [9, A.54]) A Hilbert algebra is a ∗ -algebra E endowed with a Hilbert norm || ⋅ || := √(⋅, ⋅) such that: (α) (xy, z) = (y, x∗ z) for all x, y, z ∈ E; (β) for all x ∈ E there is C > 0 such that ||xy|| ≤ C||y|| for all y ∈ E, i.e. the left multiplication maps m x : (E, || ⋅ ||) → (E, || ⋅ ||), m x (y) := xy, are bounded; (γ) (y∗ , x∗ ) = (x, y) for all x, y ∈ E; (δ) the linear span of the set E2 := {ab : a, b ∈ E} is dense in E. Each norm || ⋅ || satisfying conditions (α)–(δ) is called a Hilbert algebra norm. Remark 3.2. If E is unital, then condition (δ) in the above definition is trivially satisfied. If, moreover, E is commutative, then (α) implies (γ). Hence, every Hilbert

12 | Ciaś norm on a unital commutative ∗ -algebra satisfying condition (α) and (β) is already a Hilbert algebra norm. Definition 3.3. A Fréchet ∗ -algebra is called a DN-algebra if it admits a Hilbert dominating norm satysfying condition (α) in Definition 3.1. A DN-algebra is called a βDN-algebra if the corresponding Hilbert dominating norm satisfies conditions (α) and (β) simultaneously. Remark 3.4. In [13, Def. 1.5] M. Măntoiu and R. Purice defined a Fréchet-Hilbert algebra as a Fréchet ∗ -algebra admitting a continous Hilbert algebra norm (more precisely, in their definition the corresponding Hilbert algebra scalar product is predetermined). Hence, in view of Remark 3.2, every unital commutative βDNalgebra is a Fréchet-Hilbert algebra. Our main results read as follows. Theorem 3.5. Let E be a unital commutative Fréchet ∗ -algebra isomorphic as a Fréchet space to a nuclear power series space of infinite type. Then the following statements are equivalent. (i) E is isomorphic as a Fréchet ∗ -algebra to a complemented ∗ -subalgebra of L∗ (s). (ii) E is isomorphic as a Fréchet ∗ -algebra to a closed ∗ -subalgebra of L∗ (s). (iii) E is a DN-algebra. Theorem 3.6. Let E be a unital commutative Fréchet ∗ -algebra isomorphic as a Fréchet space to a nuclear power series space of infinite type. Then the following statements are equivalent. (i) E is isomorphic as a Fréchet ∗ -algebra to a complemented ∗ -subalgebra F of L∗ (s) such that F ⊂ L(ℓ2 ). (ii) E is isomorphic as a Fréchet ∗ -algebra to a closed ∗ -subalgebra F of L∗ (s) such that F ⊂ L(ℓ2 ). (iii) E is a βDN-algebra. We divide the proof into a sequence of lemmas. As a by-product, we obtain also three results which are interesting enough to be stated as “corollaries". For every N, n ∈ N0 we define the space L(s n , s N ) ∩ L(s−N , s−n ) := {x ∈ L(s n , s N ) : ∃x̃ ∈ L(s−N , s−n )

with the norm

r N,n (x) := max { sup |xξ|N , sup |x̃ ξ|−n }. |ξ|n ≤1

x̃|s n = x}

|ξ|−N ≤1

Formally, the space L(s n , s N ) ∩ L(s−N , s−n ) is the projective limit of the Banach spaces L(s n , s N ) and L(s−N , s−n ) with their standard norms, and thus it is a

Fréchet algebras | 13

Banach space itself. Since

sup |x∗ ξ|N = sup sup |⟨x∗ ξ, η⟩| = sup sup |⟨ξ, x̃ η⟩| = sup |x̃ η|−n ,

|ξ|n ≤1

we have

|ξ|n ≤1 |η|−N ≤1

|ξ|n ≤1 |η|−N ≤1

|η|−N ≤1

r N,n (x) = max { sup |xξ|N , sup |x∗ ξ|N }, |ξ|n ≤1

|ξ|n ≤1

where x∗ ∈ L(s n , s N ) is the hilbertian adjoint of the operator x̃. Moreover, L∗ (s) = L(s) ∩ L(s󸀠 ) (see, e.g., [8, Prop. 3.7]), hence L∗ (s) = {x : s → s : x linear and ∀N ∈ N0 ∃n ∈ N0

r N,n (x) < ∞}

as sets. Therefore, we can endow L∗ (s) with the topology of the PLB-space (a countable projective limit of a countable inductive limit of Banach spaces) projN∈N0 indn∈N0 L(s n , s N ) ∩ L(s−N , s−n ). It appears that the topology τ∗ and the PLB-topology on L∗ (s) coincide. Lemma 3.7. We have L∗ (s) = projN∈N0 indn∈N0 L(s n , s N ) ∩ L(s−N , s−n ) as topological vector spaces. Proof. By [8, Cor. 4.2], L∗ (s) is ultrabornological and projN∈N0 indn∈N0 L(s n , s N ) ∩ L(s−N , s−n ) is webbed as a PLB-space. Hence, by the open mapping theorem (see e.g. [14, Th. 24.30]), it is enough to show that the identity map ι : projN∈N0 indn∈N0 L(s n , s N ) ∩ L(s−N , s−n ) → L∗ (s)

is continuous. Let N ∈ N0 and let B be a bounded subset of s. For every m ∈ N0 choose a constant λ m > 0 such that B ⊂ {ξ ∈ s : |ξ|m ≤ λ m }. Then p N,B (x) = max { sup |xξ|N , sup |x∗ ξ|N } ≤ λ m max { sup |xξ|N , sup |x∗ ξ|N } ξ ∈B

ξ ∈B

|ξ|m ≤1

|ξ|m ≤1

= λ m r N,m (x)

for every m ∈ N0 and x ∈ L(s m , s N ) ∩ L(s−N , s−m ), and thus ι is continuous.

Lemma 3.8. For every Fréchet subspace F of L∗ (s) there is m ∈ N0 such that F ⊂ L(s m , ℓ2 ) ∩ L(ℓ2 , s−m ) and, moreover, for each such m, r m : F → [0, ∞),

is a dominating norm on F.

r m (x) := max{ sup ||xξ||ℓ2 , sup ||x∗ ξ||ℓ2 }, |ξ|m ≤1

|ξ|m ≤1

14 | Ciaś Proof. By the very definition of projective topology, the canonical embedding projN∈N0 indn∈N0 L(s n , s N ) ∩ L(s−N , s−n ) 󳨅→ indn∈N0 L(s n , ℓ2 ) ∩ L(ℓ2 , s−n ) is continuous and thus the identity map κ : F 󳨅→ indn∈N0 L(s n , ℓ2 ) ∩ L(ℓ2 , s−n ) is continuous, as well. Hence, by Grothendieck’s factorization theorem [14, Th. 24.33], there is m ∈ N such that κ(F) ⊂ L(s m , ℓ2 )∩L(ℓ2 , s−m ). Since we can identify in a obvious way F with κ(F), we get the first part of the thesis. Now, fix an arbitrary m ∈ N0 such that F ⊂ L(s m , ℓ2 ) ∩ L(ℓ2 , s−m ). Then r m is a continuous seminorm on F. Since F is a Fréchet space, there is a sequence (B N )N∈N , B N ⊂ B N+1 , of bounded subsets of s such that (p N )N∈N , p N (x) := max { sup |xξ|N , sup |x∗ ξ|N } ξ ∈B N

ξ ∈B N

for x ∈ F, is a fundamental sequence of seminorms on F. Moreover, for every N ∈ N there is λ N > 0 such that B N ⊂ {ξ ∈ s : |ξ|m ≤ λ N }. Hence, for x ∈ F and N ∈ N, we obtain p2N (x) = max { sup |xξ|2N , sup |x∗ ξ|2N } ξ ∈B N

ξ ∈B N

≤ max { sup (||xξ||ℓ2 |xξ|2N ), sup (||x∗ ξ||ℓ2 |x∗ ξ|2N )} ξ ∈B N

ξ ∈B N

≤ max { sup ||xξ||ℓ2 ⋅ sup |xξ|2N , sup ||x∗ ξ||ℓ2 ⋅ sup |x∗ ξ|2N } ξ ∈B N

ξ ∈B N

ξ ∈B N

ξ ∈B N

≤ λ N max { sup ||xξ||ℓ2 ⋅ sup |xξ|2N , sup ||x∗ ξ||ℓ2 ⋅ sup |x∗ ξ|2N }, |ξ|m ≤1

ξ ∈B2N

|ξ|m ≤1

ξ ∈B2N

where the first inequality follows from the Cauchy-Schwartz inequality. Finally, since max{ab, cd} ≤ max{a, c} ⋅ max{b, d} for all a, b, c, d ≥ 0, we obtain

p2N (x) ≤ λ N max { sup ||xξ||ℓ2 , sup ||x∗ ξ||ℓ2 } ⋅ max { sup |xξ|2N , sup |x∗ ξ|2N } |ξ|m ≤1

|ξ|m ≤1

ξ ∈B2N

ξ ∈B2N

= λ N r m (x)p2N (x)

for all x ∈ F, and thus r m is a dominating norm on F. Corollary 3.9. (i) Every Fréchet subspace of L∗ (s) is isomorphic to a closed subspace of s. (ii) Every Fréchet quotient of L∗ (s) is isomorphic to a quotient of s. (iii) Every complemented Fréchet subspace of L∗ (s) is isomorphic to a complemented subspace of s.

Fréchet algebras |

15

Proof. First note that every closed subspace and quotient of L∗ (s) is nuclear because L∗ (s) is nuclear itself (see [8, Prop. 3.8 & Cor. 4.2]). (i) This follows immediately from Lemma 3.8 and [14, Prop. 31.5]. (ii) Let E be a Fréchet quotient of L∗ (s). It follows from [8, Prop. 4.7] and [2, Cor. 1.2(a) and (c)] that E, being isomorphic to a quotient of L∗ (s), has the property (Ω). Therefore, by [14, Prop. 31.6], E is isomorphic to a quotient of s. (iii) This is a direct consequence of the previous items and [14, Prop. 31.7]. Let e j denote the j-th unit vector in CN . If F is a Fréchet subspace of L∗ (s) then, by Lemma 3.8, there is m ∈ N0 such that F ⊂ L(s m , ℓ2 ) ∩ L(ℓ2 , s−m ) and r m is a continuous (dominating) norm on F. Since, for all x ∈ F, we have [x]m := ( ∑ ||xe j ||2ℓ2 j−2m−2 ) ∞

1/2

j=1

≤ ( ∑ j−2 ) ∞

1/2

⋅ sup ||xξ||ℓ2 |ξ|m ≤1

j=1

π π ≤ max { sup ||xξ||ℓ2 , sup ||x∗ ξ||ℓ2 } = r m (x), √6 √ 6 |ξ|m ≤1 |ξ|m ≤1

the scalar product

[x, y]m := ∑ ⟨xe j , ye j ⟩j−2m−2 , ∞

[⋅, ⋅]m : F × F → C ,

(5)

j=1

is well-defined and [ ⋅ ]m = √[⋅, ⋅]m is a continuous Hilbert norm on F.

Lemma 3.10. Let F be a commutative Fréchet ∗ -subalgebra of L∗ (s) and let m ∈ N0 be such that F ⊂ L(s m , ℓ2 ) ∩ L(ℓ2 , s−m ). Then the norm [ ⋅ ]m defined by (5) is a Hilbert dominating norm on F satisfying condition (α). Proof. Since F is commutative, we have ||xξ||ℓ2 = ||x∗ ξ||ℓ2 for all x ∈ E and all ξ ∈ s. Hence,

r m+2 (x) = max { sup ||xξ||ℓ2 , sup ||x∗ ξ||ℓ2 } = sup ||xξ||ℓ2 |ξ|m+2 ≤1

|ξ|m+2 ≤1

|ξ|m+2 ≤1

󵄨󵄨 󵄨󵄨 = sup 󵄨󵄨󵄨󵄨󵄨󵄨 ∑ ξ j j m+2 ⋅ x(e j j−m−2 )󵄨󵄨󵄨󵄨󵄨󵄨ℓ2 ≤ sup ∑ |ξ j |j m+2 ⋅ ||xe j ||ℓ2 ⋅ j−m−2 ∞

|ξ|m+2 ≤1

∞

j=1

≤ ∑ ||xe j ||ℓ2 ⋅ j−m−2 ≤ ( ∑ j−2 ) ∞

∞

j=1

j=1

Therefore, [ ⋅ ]m ≥

1/2

|ξ|m+2 ≤1 j=1

⋅ ( ∑ ||xe j ||2ℓ2 j−2m−2 ) ∞

j=1

√6 r m+2 , π

1/2

=

π [x]m . √6

16 | Ciaś and, by Lemma 3.8, [⋅]m is a dominating norm on F. Moreover, we have

[xy, z]m = ∑ ⟨xye j , ze j ⟩j−2m−2 = ∑ ⟨ye j , x∗ ze j ⟩j−2m−2 = [y, x∗ z]m , ∞

∞

j=1

j=1

which completes the proof. Definition 3.11. A closed subspace E of the space s is called orthogonally complemented in s if there is a continuous projection π in s onto E admitting the extension to the orthogonal projection in ℓ2 . Then we call π an orthogonal projection in s onto E. Lemma 3.12. Let E be a Fréchet space isomorphic to a nuclear power series space of infinite type and let || ⋅ || be a dominating Hilbert norm on E. Then there is an orthogonally complemented subspace G of s and an isomorphism w : E → G of Fréchet spaces such that ||wξ||ℓ2 = ||ξ|| for all ξ ∈ E. Proof. Since E is isomorphic to a nuclear power series space of infinite type, by [14, Lemma 29.2(3) & Lemma 29.11(3)], E has the properties (DN) and (Ω). Hence, by [14, Prop. 31.7], E is isomorphic to a complemented subspace of s. This means that there is a complemented subspace F of s with a continuous projection π : s → F and a Fréchet space isomorphism ψ : E → F. Hence, || ⋅ ||ψ : F → [0, ∞) defined by ||ξ||ψ := ||ψ−1 ξ|| is a dominating Hilbert norm on F. Since, || ⋅ ||ℓ2 is also a dominating Hilbert norm on F, by [23, Cor. 7.7], there is an automorphism u of F such that ||uξ||ℓ2 = ||ξ||ψ for all ξ ∈ F. Moreover, by [23, Th. 7.2], there is an automorphism v of s such that ρ := vπv−1 is the orthogonal projection in s onto G := v(F) and a simple analysis of the proof of [23, Th. 7.2] shows that ||vξ||ℓ2 = ||ξ||ℓ2 for all ξ ∈ F. Therefore, the operator w := vuψ has the desired properties. Lemma 3.13. Let E be a Fréchet space isomorphic to a nuclear power series space of infinite type and let ||⋅|| be a dominating Hilbert norm on E. Let H denote the completion of E in the norm || ⋅ ||. Then there is a map φ ∈ L(H, ℓ2 ) and an orthogonally complemented subspace G of s such that (i) φ(E) = G; (ii) φ∗ (s) = E; (iii) ||φξ||ℓ2 = ||ξ|| for all ξ ∈ E; (iv) φϕ∗ is the orthogonal projection in ℓ2 with φϕ∗ (s) = G. Moreover, the map φ : L∗ (E, || ⋅ ||) → L∗ (s),

x 󳨃→ φxφ∗ ,

is a continuous injective ∗ -algebra homomorphism with im φ = L∗ (G) and the map P : L∗ (s) → L∗ (G),

x 󳨃→ φϕ∗ xφϕ∗ ,

Fréchet algebras |

17

is a continuous projection onto L∗ (G). Proof. By Lemma 3.12, there is an orthogonally complemented subspace G of s and an isomorphism w : E → G of Fréchet spaces such that ||wξ||ℓ2 = ||ξ|| for all ξ ∈ E. Let ρ : s → s be the orthogonal projection onto G. The operators w, ρ and the identity map ι : G 󳨅→ s can be extended to the continuous linear operators between Hilbert spaces (for simplicity denoted by the same symbols): w : H → G, ρ : ℓ2 → ℓ2 and ι : G 󳨅→ ℓ2 , where G is the closure of G in ℓ2 . Therefore, the Hermitian adjoints w∗ and ι∗ of the operators w and ι are well-defined. We have thus the following commutative diagram of continuous linear maps between Fréchet and Hilbert spaces E

w

G

ι

s

H

w

G

ι

ℓ2

and the diagram with the corresponding adjoint operators ℓ2

ι∗

G

ρ

w∗

H

ι

ℓ2 . It follows easily that ι∗ : ℓ2 → G is the orthogonal projection onto G, whence = ρ(s) = G. Moreover, w∗ (G) = E. Indeed, if (⋅, ⋅) denotes the scalar product on E corresponding to the Hilbert norm || ⋅ ||, then ι∗ (s)

(w∗ wξ, η) = ⟨wξ, wη⟩ = (ξ, η) for all ξ, η ∈ E. Hence, E being dense in H, w∗ w = idH , and so w∗ (G) = E. Consequently, we have the following commutative diagram s

ι∗

G ρ

w∗

E

ι

s. It is easy to check that φ := ιw satisfies conditions (i)–(iii) and a simple computation shows that φϕ∗ is a self-adjoint projection (and thus orthogonal) on ℓ2 with φϕ∗ (s) = G. In consequence, φ : L∗ (E, || ⋅ ||) → L∗ (s), x 󳨃→ φxφ∗ , is an injective ∗ -homomorphism with im φ = L∗ (G) and, moreover, P : L∗ (s) → L∗ (G), x 󳨃→ φϕ∗ xφϕ∗

18 | Ciaś is a projection. Now, we shall prove the continuity of φ. Let B be a bounded subset of s and let n ∈ N0 . By the closed graph theorem, φ : E → s and φ∗ : s → E are continuous maps between Fréchet spaces. Hence, there is a constant C > 0 and a continuous norm || ⋅ ||E on E such that |φξ|n ≤ C||ξ||E for ξ ∈ E. Note also that the set φ∗ (B) is bounded in the Fréchet space E. Therefore, max{|(φx)ξ|n , |(φx)∗ ξ|n } = max{|φxφ∗ ξ|n , |φx∗ φ∗ ξ|n } ≤ C max {||xη||E , ||x∗ η||E }, ∗ ξ ∈B

ξ ∈B

η∈φ (B)

which, by Proposition 2.6, gives the continuity of φ. The continuity of P can be proved in a simillar way. Corollary 3.14. For every Fréchet space E isomorphic to a nuclear power series space of infinite type and a dominating Hilbert norm || ⋅ || on E there is an orthogonally complemented subspace G of s such that L∗ (E, || ⋅ ||) ≅ L∗ (G) as topological ∗ -algebras. Moreover, the algebra L∗ (G) is a complemented ∗ -subalgebra of L∗ (s). Proof. This follows directly from Lemma 3.13. In our next corollary we deal with PLS-spaces, i.e. countable projective limits of strong duals of Fréchet-Schwartz spaces (see [10] for basic properties and examples). Corollary 3.15. Let E be a Fréchet space isomorphic to a nuclear power series space of infinite type and let || ⋅ || be a dominating Hilbert norm on E. Then the space L∗ (E, || ⋅ ||) is a nuclear, ultrabornological PLS-space. Proof. By [8, Prop. 3.8 & Cor. 4.2], L∗ (s) is a nuclear, ultrabornological PLS-space. These properties are inherited by complemented subspaces (see [14, Prop. 28.6], [17, Ch. II, 8.2, Cor. 1], [11, Prop. 1.2]). Hence, the desired conclusion follows from Corollary 3.14. Lemma 3.16. Let E be a unital DN-algebra isomorphic as a Fréchet space to a nuclear power series space of infinite type and let || ⋅ || := √(⋅, ⋅) be the corresponding Hilbert norm. Let ME := {m x : x ∈ E}, where m x : E → E, m x y := xy, denotes the left multiplication map for the element x. Then ME is a complemented ∗ -subalgebra of L∗ (E, || ⋅ ||) and E is isomorphic as a Fréchet ∗ -algebra to ME .

Proof. By assumption, (m x y, z) = (xy, z) = (y, x∗ z) = (y, m x∗ z)

Fréchet algebras |

19

for all x, y, z ∈ E, hence E ⊂ D((m x )∗ ) and (m x )∗ |E = m x∗ . Consequently, m x ∈ L∗ (E, || ⋅ ||) for all x ∈ E. Define Q : L∗ (E, || ⋅ ||) → L∗ (E, || ⋅ ||) by Qφ := m φ(1) , where 1 is the unit in E. Clearly, Q is a projection onto ME ; we will show that Q is continuous. By Corollary 3.15 and the closed graph theorem (see e.g. [14, Th. 24.31]), every linear map on L∗ (E, || ⋅ ||) with closed graph is continuous. Assume that a net (φ λ )λ ⊂ L∗ (E, || ⋅ ||) converges to 0, (Qφ λ )λ converges to ψ and both limits are taken in L∗ (E, || ⋅ ||). Let us fix x ∈ E. By the continuity of the multiplication in E, there is C > 0 and a continuous norm || ⋅ ||1 on E with ||yx|| ≤ C||y||1 for all y ∈ E. Hence, we have ||ψx|| ≤ ||(ψ − Qφ λ )x|| + ||Qφ λ x|| = ||(ψ − Qφ λ )x|| + ||φ λ (1)x|| ≤ ||(ψ − Qφ λ )x|| + C||φ λ (1)||1 .

By assumption, ||(ψ − Qφ λ )x|| → 0 and ||φ λ (1)||1 → 0, which yields ψx = 0. Consequently, ψ = 0 and Q is continuous. Finally, we should show that E is isomorphic as a topological ∗ - algebra to ME – a complemented ∗ -subalgebra of L∗ (E, || ⋅ ||). Let us consider the map φ : E → ME , φx := m x . By the above, it is clear that φ is a ∗ -algebra isomorphism. Let B be a bounded subset of E and let || ⋅ ||0 be a continuous norm on E. Since the multiplication on E is jointly continuous, there is C1 > 0 and a continuous norm || ⋅ ||1 on E such that ||xy||0 ≤ C1 ||x||1 ||y||1 for x, y ∈ E. Moreover, by the continuity of the involution, there is a constant C2 ≥ 1 and a continuous norm || ⋅ ||2 on E such that || ⋅ ||2 ≥ || ⋅ ||1 and ||x∗ ||1 ≤ C2 ||x||2 for x ∈ E. Hence, max{sup ||m x y||0 , sup ||(m x )∗ y||0 } = max{sup ||xy||0 , sup ||x∗ y||0 } y∈B

y∈B

y∈B

y∈B

≤ C1 sup ||y||1 max{||x||1 , ||x ||1 } ≤ C1 C2 C3 ||x||2 , ∗

y∈B

where C3 := supy∈B ||y||1 < ∞. This shows that φ is continuous. Since E and ME (as a complemented subspace of L∗ (E, || ⋅ ||), see also Corollary 3.15) satisfy assumptions of the open mapping theorem [14, Th. 24.30], the map φ is an isomorphism of Fréchet ∗ -algebras, which completes the proof. Proof of Theorem 3.5. The implication (i)⇒(ii) is trivial. (ii)⇒(iii): Let F be a closed ∗ -subalgebra of L∗ (s) such that E ≅ F as Fréchet ∗ algebras and let T : E → F be the corresponding isomorphism. By Lemma 3.8, there is m ∈ N0 such that F ⊂ L(s m , ℓ2 ) ∩ L(ℓ2 , s−m ) and thus, by Lemma 3.10, [ ⋅ ]m = √[⋅, ⋅]m is a dominating Hilbert norm on F such that [xy, z]m = [y, x∗ z]m for all x, y, z ∈ F. Let (⋅, ⋅) : E × E → C, (x, y) := [Tx, Ty]m . Then, clearly, (xy, z) = (y, x∗ z) for all x, y, z ∈ E and || ⋅ || := √(⋅, ⋅) is a dominating Hilbert norm on E, hence E is a DN-algebra.

20 | Ciaś (iii)⇒(i): By Lemma 3.16, E is isomorphic to a complemented ∗ -subalgebra of L∗ (E, || ⋅ ||) and, by Corollary 3.14, L∗ (E, || ⋅ ||) is isomorphic to a complemented ∗ -subalgebra of L∗ (s), which proves the theorem. ◻ Proof of Theorem 3.6. Clearly, (i) implies (ii). (ii)⇒(iii): Let φ : E → F be the isomorphism of the Fréchet ∗ -algebras E and F. Since F ⊂ L(ℓ2 ), by Lemma 3.10, [ ⋅ ]0 is a Hilbert dominating norm on F satisfying condition (α). Consequently, || ⋅ || := [φ(⋅)]0 is a Hilbert dominating norm on E and it satisfies condition (α). Next, for all x ∈ E there is C > 0 such that ||(φ(xy))e j ||ℓ2 = ||(φx)((φy)e j )||ℓ2 ≤ C||(φy)e j ||ℓ2 for all y ∈ E and j ∈ N. Hence, for all x ∈ E there is C > 0 such that ||xy|| = [φ(xy)]0 = ( ∑ ||(φ(xy))e j ||2ℓ2 j−2 ) ∞

j=1

= C[φy]0 = C||y||

1/2

≤ C( ∑ ||(φy)e j ||2ℓ2 j−2 ) ∞

1/2

j=1

for all y ∈ E, which gives condition (β) in Definition 3.1. This shows that E is a βDN-algebra. (iii)⇒(i): Let H be the completion of E in the norm ||D|| and let ME := {m x : x ∈ E}, where m x : E → E, m x (y) := xy. By Lemma 3.16, ME is a complemented ∗ subalgebra of L∗ (E, || ⋅ ||) isomorphic to E. Moreover, by Lemma 3.13, L∗ (E, || ⋅ ||) is isomorphic to a complemented ∗ -subalgebra of L∗ (s) via the map φ : L∗ (E, || ⋅ ||) → L∗ (s),

x 󳨃→ φxφ∗ ,

where φ ∈ L(H, ℓ2 ) and its adjoint φ∗ ∈ L(ℓ2 , H) satisfy conditions (i)–(iv) in Lemma 3.13. Hence, the assigment x 󳨃→ φm x φ∗ defines an isomorphism (of Fréchet ∗ -algebras) between E and a complemented ∗ -subalgebra of L∗ (s). Now, it is left to show that φm x φ∗ ∈ L(ℓ2 ) for each x ∈ E. By assumption, for each x ∈ E, the map m x : (E, || ⋅ ||) → (E, || ⋅ ||) is continuous. Consequently, for each x ∈ E there are C1 , C2 > 0 such that ||φm x φ∗ ξ||ℓ2 = ||m x φ∗ ξ|| ≤ C1 ||φ∗ ξ|| ≤ C2 ||ξ||ℓ2 for all ξ ∈ ℓ2 , and the proof is complete.

◻

Remark 3.17. The proofs of implication (iii)⇒(i) in Theorems 3.5 and 3.6 work also when E is not commutative. The implication (ii)⇒(iii) in Theorems 3.5 and 3.6 holds for any commutative Fréchet ∗ -algebra.

Fréchet algebras |

21

4 Examples In this section we present several examples of classes of commutative Fréchet ∗ algebras which can be embedded into L∗ (s) as complemented ∗ -subalgebras consisting of bounded operators on ℓ2 . In the case of unital algebras, in view of Theorem 3.6, it is enough to show that a given Fréchet ∗ -algebra is isomorphic to a complemented subspace of s with Schauder basis (i.e. to a nuclear power series space of infinite type) and admits a Hilbert dominating norm satisfying conditions (α) and (β) in Definition 3.1. Nonunital algebras will be extended in a natural way to unital ones . At the end of this section we also give one interesting counterexample. By E(K) we denote the space of (complex-valued) Whitney jets on a compact set K ⊂ R n , E(K) := {(∂ α F |K )α∈N0n : F ∈ C∞ (R n )}. The space E(K) thus consists of some special sequences f = (f (α) )α∈Nn of continous 0 functions on the set K. The Fréchet space topology on E(K) is given by the system of seminorms (|| ⋅ ||m )m∈N defined, for example, in [12, Section 2]; here let us only note that sup{|f (α) (x)| : x ∈ K, |α| ≤ m} ≤ ||f||m for all m ∈ N0 and f ∈ E(K). The space E(K) is a Fréchet ∗ -algebra where the product fg of f, g ∈ E(K) is defined by the Leibniz rule, i.e., α (fg)(α) := ∑ ( )f (β) g (α−β) β β≤α

for α ∈ N0n (see also [12, p. 133]). As involution we clearly take the pointwise conjugation, f := (f (α) )α∈Nn . We say that a compact set K ⊂ R n has the extension 0 property if there exists a continuous linear operator E : E(K) → C∞ (R n ) such that ∂ α (Ef)|K = f (α) for every α ∈ N0n . M. Tidten showed in [20, Folgerung 2.4] that a compact set K ⊂ R n has the extension property if and only if E(K) has the property (DN). All ∗ -algebras of smooth functions considered below are endowed with pointwise multiplication and conjugation. The algebra H(C) of entire functions is endowed with pointwise multiplication and the involution f 󳨃→ f ∗ defined by f ∗ (z) := f(z). Let K∞ := {(x ij )i,j∈N ∈ CN : sup |x ij |(ij)n < ∞ 2

i,j∈N

for all n ∈ N0 }

22 | Ciaś be the algebra of rapidly decreasing matrices endowed with matrix multiplication and conjugation of the transpose as involution. The algebra K∞ is isomorphic as the Fréchet ∗ -algebra to the algebra L(s󸀠 , s) of compact smooth operators. Moreover, it is isomorphic as a Fréchet space to the space s. For further information concerning the algebra K∞ we refer the reader to [5, 6, 7]. Theorem 4.1. The following Fréchet ∗ -algebras are isomorphic to some complemented ∗ -subalgebra of L∗ (s) consisting of bounded operators on ℓ2 . (i) The algebras C∞ (M) of smooth functions on compact second-countable smooth manifolds M without boundary. (ii) The algebras D(K) of smooth functions on R n with support contained in compact sets K ⊂ R n such that int K ≠ 0. (iii) The algebra S(R n ) of smooth rapidly decreasing functions on R n . (iv) The algebras E(K) of Whitney jets on compact sets K ⊂ R n with the extension property admitting a Schauder basis. (v) The algebra H(C) of entire functions. (vi) Nuclear power series algebras Λ∞ (α) of infinite type with pointwise multiplication and conjugation. (vii)The algebra K∞ of rapidly decreasing matrices. Proof. (i) We follow the general pattern of the reasoning of P. Michor [15]. Let us choose a Riemannian metric g on M and let dV be the volume element (density) on (M, g). Let ⟨⋅, ⋅⟩L2 (M) be the scalar product on L2 (M) defined by ⟨f, g⟩L2 (M) := ∫ f gdV M

and let || ⋅ ||L2 (M) denote the corresponding Hilbert norm on L2 (M). By the SturmLiouville decomposition [3, pp. 139–140], eigenfunctions (u k )k∈N of the Laplacian ∆ induced by g form an orthonormal basis of L2 (M) and the sequence (λ k )k∈N of eigenvalues satisfies 0 = λ1 < λ2 ≤ λ3 ≤ . . . and λ k → ∞. Moreover, by the Weyl asymptotic formula [4, Note III.15, p. 184], there is a constant C > 0 depending only on n and the choice of a Riemannian metric such that 2 (6) λ k ∼ Ck n , as k → ∞. We claim that (u k )k∈N is a Schauder basis of C∞ (M) whose coefficient space is equal to the space s of rapidly decreasing sequences. Indeed, for each r ∈ N, the operator (I + ∆)r : H 2r (M) → L2 (M)

Fréchet algebras | 23

is an isomorphism between the Sobolev space H 2r (M) and L2 (M). Therefore, since (I + ∆)r u k = (1 + λ k )r u k , we have

u k = (1 + λ k )r (I + ∆)−r u k ∈ H 2r (M)

for all k, r ∈ N. Consequently each u k belongs to ⋂r∈N H 2r (M) = C∞ (M). Next, since (u k )k∈N is an orthonormal basis of L2 (M), for each f ∈ C∞ (M) one can find a unique sequence (a k )k∈N of scalars such that f = ∑k=1 a k u k and the series converges in the norm ||⋅||L2 (M) . In particular, for each r ∈ N there is a unique sequence (a k,r )k∈N ⊂ CN such that (I + ∆)r f = ∑ a k,r u k . k=1

Since (I + ∆)r is a symmetric unbounded operator on L2 (M), we have

a k,r = ⟨(I + ∆)r f, u k ⟩L2 (M) = ⟨f, (I + ∆)r u k ⟩L2 (M) = ⟨f, (1 + λ k )r u k ⟩L2 (M) = a k (1 + λ k )r , whence

(I + ∆)r f = ∑ a k (1 + λ k )r u k k=1

and the series converges in the norm || ⋅ ||L2 (M) . Therefore, (a k (1 + λ k )r )k∈N ∈ ℓ2 2 for all r ∈ N and, by the Weyl asymptotic formula (6), (a k (1 + Ck n )r )k∈N ∈ ℓ2 for all r ∈ N, which yields (a k )k∈N ∈ s. Finally, it is a simple matter to show that for each (a k )k∈N ∈ s the series ∑k=1 a k u k converges in C∞ (M). Hence, (u k )k∈N is a Schauder basis of C∞ (M) with the coefficient space equal to s as claimed. Now, T : C∞ (M) → s defined by Tu k = e k (e k denotes the k-th unit vector in CN ) is an isomorphism of Fréchet spaces such that ||Tf||ℓ2 = ||f||L2 (M) for f ∈ C∞ (M). Therefore, since ||⋅||ℓ2 is a dominating norm on s, ||⋅||L2 (M) is a dominating norm on C∞ (M). Clearly, ⟨fg, h⟩L2 (M) = ⟨g, f h⟩L2 (M)

and

||fg||L2 (M) ≤ sup |f(x)| ⋅ ||g||L2 (M) x∈M

for all f, g, h ∈ C∞ (M). Hence, by Theorem 3.6, C∞ (M) is isomorphic to a complemented ∗ -subalgebra of L∗ (s) consisting of bounded operators on ℓ2 . (ii) Choose a ∈ R n and r > 0 such that K is contained in the open ball B(a, r) of radius r centered at a. Then, clearly, D(K) is isomorphic as a Fréchet ∗ -algebra to the algebra D(K, B(a, r)) := {f ∈ C∞ (B(a, r)) : supp(f) ⊂ K} which is a closed subspace of the Fréchet space C∞ (B(a, r)) of smooth functions on B(a, r) with uniformly continuous partial derivatives (see [14, Ex. 28.9(5)]). Let

24 | Ciaś 1 be the constant function on B(a, r), everywhere equals 1. Let D1 (K, B(a, r)) denote the linear span of D(K, B(a, r)) and 1. Then, clearly, 1 is the unit in the Fréchet ∗ -algebra D1 (K, B(a, r)). By [14, Prop. 31.12], D(K, B(a, r)) is isomorphic as a Fréchet space to s, and so is D1 (K, B(a, r)). Now, we follow the proof of [14, Lemma 31.10]. By [14, Prop. 14.27], (|| ⋅ ||m )m∈N , ||f||2m := ∑

∫ |f (α) (x)|2 dx = ∑ ||f (α) ||2L2 (B(a,r)) ,

|α|≤m B(a,r)

|α|≤m

is a fundamental sequence of norms on C∞ (B(a, r)), and thus on D1 (K, B(a, r)). Since (f + λ1)(α) = f (α) for |α| > 0, we have, by integration by parts and by the Cauchy-Schwartz inequality, ||(f + λ1)(α) ||2L2 (B(a,r)) = ∫ |(f + λ1)(α) |2 dx = ∫ (f + λ1)(α) (f + λ1)(α) dx B(a,r)

= (−1)

|α|

B(a,r)

∫ (f + λ1)(f + λ1)(2α) dx

B(a,r)

≤ ||f + λ1||L2 (B(a,r)) ||(f + λ1)(2α) ||L2 (B(a,r)) for all f + λ1 ∈ D1 (K, B(a, r)) and |α| ≤ m. Hence, for all m ∈ N0 there is a constant C m > 0 such that ||f + λ1||2m ≤ C m ||f + λ1||L2 (B(a,r)) ||f + λ1||2m for all f + λ1 ∈ D1 (K, B(a, r)). Hence, || ⋅ ||L2 (B(a,r)) is a dominating norm on D1 (K, B(a, r)), and the corresponding scalar product satisfies condition (α) in Definition 3.1. Futhermore, ||(f + λ1)(g + μ1)||L2 (B(a,r)) = ( ∫ |f(x) + λ|2 |g(x) + μ|2 dx)

1/2

B(a,r)

≤ sup |f(x) + λ| ⋅ ||g + μ1||L2 (B(a,r)) . x∈B(a,r)

Hence, D1 (K, B(a, r)) is a unital βDN-algebra. By Theorem 3.6, D1 (K, B(a, r)) is isomorphic to a complemented ∗ -subalgebra of L∗ (s) consisting of bounded operators on ℓ2 , and so is D(K). (iii) It is well-known that the map φ : S(R n ) → ⋅([ −

π π , ]), 2 2

(φf)(x1 , . . . , x n ) := (tan x1 , . . . , tan x2 ),

Fréchet algebras | 25

is an isomorphism of Fréchet ∗ -algebras (see [14, Ex. 29.5(3) and p. 402]), hence the conclusion follows from the previous example. Moreover, || ⋅ || : S(R n ) ⊕ C1 → [0, ∞), π 2

π 2

− 2π

− 2π

||f + λ1||2 : = ∫ . . . ∫ |f(tan x1 , . . . , tan x2 ) + λ|2 dx1 . . . dx n , is a Hilbert dominating norm on S(R n ) ⊕ C1 satisfying conditions (α) and (β). (iv) We shall show that there is a finite positive Borel measure μ on K such that || ⋅ ||L2 (μ) , ||f||2L2 (μ) := ∫K |f|2 dμ, is a dominating norm on E(K). Then, since || ⋅ ||L2 (μ) is a Hilbert algebra norm, we would get our conclusion. By [12, Th. 3.10], the norm | ⋅ |0 , |f|0 := supx∈K |f(x)|, is a dominating norm on E(K). This means that for all k there is l and C > 0 such that ||f||k ≤ C||f||l |f|0 1/2

1/2

(7)

for all f ∈ E(K). If there were a finite positive Borel measure μ on K, q ∈ N and C > 0 such that 1/2 1/2 |f|0 ≤ C||f||q ||f||L2 (μ) (8) for all f ∈ E(K) then, by (7), we would get 1

1

3

1

4 ||f||k ≤ C||f||l ||f||q4 ||f||L42 (μ) ≤ C||f||max{l,q} ||f||L42 (μ) ,

1/2

and hence || ⋅ ||L2 (μ) would be a dominating norm on E(K) (see [12, Remark 3.2(ii)]). Our goal is thus to prove condition (8). In what follows, C ≥ 1 denotes a constant which can vary from line to line but depends only on n and the set K. By the last line of the proof of [1, Prop. 3.4], there is a positive Borel measure μ on K – the so called Bernstein-Markov measure – such that n+j 2 |p|0 ≤ C( (9) )j ||p||L2 (μ) ≤ Cj m ||p||L2 (μ) . j for all polynomials p ∈ C[x1 , . . . , x n ] with deg(p) ≤ j and where m := n + 2. Let us fix f ∈ E(K). By [12, Cor. 4.4(i)], for all j ∈ N there is a polynomial p j with deg(p j ) ≤ j such that |f − p j |0 ≤ Cj−2m ||f||2m . Applying this inequality twice and also inequality (9), we obtain |f|0 ≤ |f − p j |0 + |p j |0 ≤ C(j−2m ||f||2m + |p j |0 ) ≤ C(j−2m ||f||2m + j m ||p j ||L2 (μ) )

≤ C[j−2m ||f||2m + j m (|f − p j |0 + ||f||L2 (μ) )] ≤ C(j−2m ||f||2m + j−m ||f||2m + j m ||f||L2 (μ) )

≤ C(j−m ||f||2m + j m ||f||L2 (μ) )

26 | Ciaś for all j ∈ N. Therefore, taking the infimum over j ∈ N in both sides of the above chain of inequalities and applying [12, Lemma 4.5], we obtain |f|0 ≤ C||f||2m ||f||L2 (μ) 1/2

1/2

for all f ∈ E(K), and (8) holds, as desired. (v) We will show that the norm || ⋅ ||L2 [−1,1] , which obviously satisfies conditions (α) and (β) in Definition 3.1, is at the same time a dominating norm. Let us recall that the Fréchet space topology on H(C) is given by the sequence of norms (|| ⋅ ||r )r∈N , ||f||r := sup|z|=r |f(z)|. By Hadamard’s three circle theorem, ||f||2r ≤ ||f||1 ||f||r2

(10)

for all r > 1 and all f ∈ H(C \ {0}), where H(C \ {0}) is the space of holomorphic functions on C \ {0}. Let us consider the map Ψ : C \ {0} → C, Ψ(z) := 21 (z + z−1 ) and let E r := Ψ(T r ), where T r := {z ∈ C : |z| = r} for r > 0. For r > 1, one can show that E r is the ellipse with the semi-axes a = 12 (r + r−1 ) and b = 21 (r − r−1 ). Moreover, E1 = [−1, 1] and E r−1 = E r for r > 0. Since f ∘ Ψ ∈ H(C \ {0}) for all f ∈ H(C), we have, by (10), ||f ∘ Ψ||2r ≤ ||f ∘ Ψ||1 ||f ∘ Ψ||r2 . Hence,

||f||2E r ≤ ||f||[−1,1] ||f||E r2

for all r > 1 and all f ∈ H(C), where ||f||E is the supremum norm on a subset E of C. Since the ellipses E r contain arbitrary big circles, || ⋅ ||[−1,1] is a dominating norm on H(C). Let (Q k )k∈N0 be the sequence of Legendre polynomials, Q k (x) := (

2k + 1 1/2 1 dk 2 (x − 1)k ) 2 2k k! dx k

for x ∈ [−1, 1]. Then, for each n ∈ N, (Q k )0≤k≤n is an orthonormal basis of the space Pn of complex-valued polynomials of degree at most n in one real variable 1 endowed with the scalar product ⟨p, q⟩ := ∫−1 p(x)q(x)dx. It is well-known that |Q k (x)| ≤ (

2k + 1 1/2 ) 2

for all x ∈ [−1, 1]. Therefore, if p ∈ Pn , p = ∑nk=0 c k Q k for some scalars c k ’s, then |p(x)| ≤ ( ∑ |Q k (x)| ) n

k=0

1/2

2

( ∑ |c j | ) n

k=0

1/2

2

≤ (n + 1)||p||L2 [−1,1]

Fréchet algebras |

27

for every x ∈ [−1, 1]. Consequently, ||p||[−1,1] ≤ (n + 1)||p||L2 [−1,1]

(11)

for every p ∈ Pn . This shows that the Lebesgue measure is a Bernstein-Markov measure on the interval [−1, 1]. n j j Now, let us take f ∈ H(C), f = ∑∞ j=0 a j z . Let p n := ∑j=0 a j z for n ∈ N0 . Then, for every n ∈ N0 and x ∈ [−1, 1], we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 ∞ |f(x) − p n (x)| = 󵄨󵄨󵄨󵄨 ∑ a j x j 󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨 ∑ a j x j−n−1 󵄨󵄨󵄨󵄨 ⋅ |x|n+1 󵄨 󵄨 󵄨 j=n+1 󵄨 j=n+1 ≤

=

󵄨󵄨 1 󵄨󵄨󵄨 ∞ 1 󵄨󵄨 ∑ (j + n + 1) . . . (j + 1)a j+n+1 x j 󵄨󵄨󵄨 = |f (n+1) (x)| 󵄨 󵄨 (n + 1)! 󵄨 j=0 󵄨 (n + 1)! 1 󵄨󵄨󵄨 󵄨󵄨 2π 󵄨󵄨

∫

|z−x|=2

󵄨󵄨 f(z) 󵄨󵄨 ≤ 1 sup |f(z)| ≤ 1 sup |f(z)|. dz 󵄨 (z − x)n+2 󵄨󵄨 2n+1 |z−x|=2 2n+1 |z|≤3

Hence, ||f − p n ||[−1,1] ≤

1 ||f||3 . 2n+1

(12)

Now, by applying computations from item (iv), we may derive from inequalities (11) and (12) that || ⋅ ||L2 [−1,1] is a dominating norm on H(C) as claimed. (vi) Let us define || ⋅ || : Λ∞ (α) ⊕ C1 → [0, ∞),

||x + λ||2 := ∑ |x j + λ|2 j−2 ∞

j=1

and || ⋅ ||t : Λ∞ (α) ⊕ C1 → [0, ∞),

||x + λ||t := max{sup |x j |e tα j , |λ|} j∈N

for t ∈ N, where 1 := (1, 1, . . .). By nuclearity, (||⋅||t )t∈N is a fundamental sequence of norms on Λ∞ (α) ⊕ C1 and || ⋅ || is a well-defined continuous Hilbert norm on Λ∞ (α) ⊕ C1. Let (⋅, ⋅) denote the scalar product corresponding to the norm || ⋅ ||. An elementary computation shows that ((x + λ)(y + μ), z + ν) = ((y + μ), (x + λ)(z + ν)) for all x + λ, y + μ, z + ν ∈ Λ∞ (α) ⊕ C1. Moreover, we have ||(x + λ)(y + μ)|| = ( ∑ |(x j + λ)(y j + μ)|2 j−2 ) ∞

j=1

1/2

≤ sup |x j + λ| ||y + μ|| j∈N

28 | Ciaś for all x + λ, y + μ ∈ Λ∞ (α) ⊕ C1. Hence, the norm || ⋅ || satisfies conditions (α) and (β) in Definition 3.1. We will show that || ⋅ || is a dominating norm on Λ∞ (α) ⊕ C1, i.e. ∀s ∈ N ∃t ∈ N ∃C > 0 ∀x + λ ∈ Λ∞ (α) ⊕ C1 ||x + λ||2s ≤ C||x + λ|| ⋅ ||x + λ||t . (13) Fix s ∈ N, x + λ ∈ Λ∞ (α) ⊕ C1 and denote R(t) := ||x + λ|| ⋅ ||x + λ||t for t ∈ N. Next, note that

1 log j log(j + 1) log(ej2 ) ≤ = +2 αj αj αj αj

for every j ∈ N and, by the nuclearity of Λ∞ (α), there is a constant γ ∈ N such log j log(j+1) 1 ≤ γ, and thus α j + 2 α j ≤ γ for all j ∈ N. Hence, αj

for all j ∈ N. We first claim that

e γα j ≥1 j+1 |λ|2 ≤ 4R(γ).

(14)

Indeed, since |x j + λ| → |λ| as j → ∞, the set A := {j ∈ N : |x j + λ| < or even empty. If A = 0, i.e. |x j + λ| ≥ |λ| 2 for all j ∈ N, then R(γ) ≥ |x1 + λ|

||x + λ|| ≥ |x j0 +1 + λ|(j0 + 1)−1 ≥ |λ| 2

implies that |x j0 | >

|λ| 2 ,

|λ| 2 }. Then, clearly,

|λ| (j0 + 1)−1 . 2

and thus

||x + λ||t ≥ |x j0 |e tα j0 >

|λ| tα j e 0 2

for all t ∈ N. Consequently, R(γ) ≥ as claimed.

is finite

|λ| |λ|2 ≥ . 2 2

Now, assume that A ≠ 0 and let j0 := max{j ∈ N : |x j + λ| < |x j0 +1 + λ| ≥ |λ| 2 , which gives

Next, |x j0 + λ|
2|λ|}. First take k ∈ A1 . Then |x k + λ| ≥

|λ| 2 ,

and so

e tα k e tα k e tα k e γα k |λ| |x k | ≥ |x k |2 ≥ |x k |2 = |x k |2 e(t−γ)α k 2 k k k+1 k+1 2 2sα k . (15) = |x k | e

R(t) ≥ |x k + λ|k−1 |x k |e tα k ≥ ≥ |x k |2 e(t−γ)α k

Now, take k ∈ A2 . Clearly the set A2 is finite. Let k0 := max{j ∈ N : |x j | > 2λ }. |λ| Then |x k0 +1 | ≤ |λ| 2 , and thus |x k0 +1 + λ| ≥ 2 . Consequently, we get |λ| |λ|2 e tα k0 |λ| (k0 + 1)−1 e tα k0 = 2 2 4 k0 + 1 |x k |2 2sα k e γα k0 ≥ e . (16) k0 + 1 16

R(t) ≥ |x k0 +1 + λ|(k0 + 1)−1 |x k0 |e tα k0 ≥ ≥

|x k |2 e tα k0 |x k |2 (t−γ)α k 0 = e 16 k0 + 1 16

Finally, fix k ∈ A3 . Then |x k + λ| > R(t) ≥ |x k + λ|k−1 |x k |e tα k ≥

|x k | 2 ,

and thus

|x k |2 (t−γ)α k e γα k |x k |2 2sα k |x k |2 e tα k . ≥ e ≥ e 2 k 2 k+1 2

(17)

Combining (15)–(17), we get sup |x j |2 e2sα j ≤ 16R(2s + γ) j∈N

as claimed. Therefore, by (14), ||x + λ||2s ≤ 16||x + λ|| ⋅ ||x + λ||2s+γ , for all x + λ ∈ Λ∞ (α) ⊕ C1, and || ⋅ || is a dominating norm on Λ∞ (α) ⊕ C1. Consequently, (Λ∞ (α) ⊕ C1, (⋅, ⋅)) is a βDN-algebra. Finally, by the above, Λ∞ (α) ⊕ C1 has the property (DN) and, by [14, Ex. 1, Ch. 29], it has the property (Ω). Moreover, by [14, Prop. 28.7], the space Λ∞ (α) ⊕ C1 is nuclear. Consequently, by [14, Prop. 31.7], Λ∞ (α) ⊕ C1 is isomorphic to a complemented subspace of s with Schauder basis, i.e. to a nuclear power series space of infinite type. Now, the thesis follows from Theorem 3.6 and from a simple observation that Λ∞ (α) is a complemented ∗ -subalgebra of Λ∞ (α) ⊕ C1.

30 | Ciaś 2

(vii) Let 1 be the identity matrix in CN and let e j be the j-th unit vector in CN . We define on K∞ ⊕ C1 the Hilbert norm || ⋅ || by ||x + λ||2 := ( ∑ ||(x + λ)e j ||2ℓ2 j−2 )

1/2

.

j∈N

Clearly, || ⋅ || satisfies conditions (α) and (β) in Definition 3.1. Hence, by Theorem 3.6 and Remark 3.17, it is enough to show that || ⋅ || is a dominating norm, i.e. ∀k ∈ N ∃n ∈ N ∃C > 0 ∀x + λ ∈ K∞ ⊕ C1 ||x + λ||2k ≤ C||x + λ|| ⋅ ||x + λ||n , where

||x + λ||k := max{sup |x ij |(ij)k , |λ|}. i,j∈N

Let us fix x + λ ∈ K∞ ⊕ C1. Note that

||(x + λ)e j ||2ℓ2 = ||(x ij )i∈N + λe j ||2ℓ2 = ∑ |x ij |2 + |x jj + λ|2 i∈N\{j}

for every j ∈ N, and thus

||x + λ||2 = ∑ ( ∑ |x ij |2 + |x jj + λ|2 )j−2 . j∈N

i∈N\{j}

We have sup |x ij |2 (ij)2k ≤ sup |x ij |j−1 ⋅ sup |x ij |(ij)2k+1

i,j∈N,i=j̸

≤(

i,j∈N,i=j̸

∑

|x ij |2 j−2 )

1/2

i,j∈N,i=j̸

i,j∈N

⋅ sup |x ij |(ij)2k+1 i,j∈N

≤ ( ∑ ( ∑ |x ij |2 + |x jj + λ|2 )j−2 )

1/2

j∈N

⋅ max{sup |x ij |(ij)2k+1 , |λ|} i,j∈N

i∈N\{j}

= ||x + λ|| ⋅ ||x + λ||2k+1

for all k ∈ N. Moreover, from (13) applied to the algebra s ⊕ C1, it follows that for every k ∈ N there is m ∈ N and C > 0 such that max{sup|x jj |2 j4k , |λ|2 } ≤ C( ∑ |x jj + λ|2 j−2 )

1/2

j∈N

max{sup |x jj |j2m , |λ|} j∈N

j∈N

≤ C( ∑ ( ∑ |x ij |2 + |x jj + λ|2 )j−2 )

1/2

j∈N

i∈N\{j}

= C||x + λ|| ⋅ ||x + λ||m .

⋅ max{sup |x ij |(ij)m , |λ|} i,j∈N

Fréchet algebras | 31

Therefore, for all k ∈ N there is n ∈ N and C > 0 such that

||x+λ||2k = C max { sup |x ij |2 (ij)2k , max { sup |x jj |2 j4k , |λ|2 }} ≤ C||x+λ||⋅||x+λ||n , i,j∈N,i=j̸

j∈N

and thus || ⋅ || is a dominating norm on K∞ ⊕ C1, which completes the proof.

Remark 4.2. (i) Every algebra D(K) is a closed ∗ -subalgebra of E(L) for any closed ball L containing K and thus, by Theorem 4.1(ii), D(K) is automatically isomorphic to a closed ∗ -subalgebra of L∗ (s) consisting of bounded operators on ℓ2 . In Theorem 4.1(iv), we prove that such a ∗ -subalgebra can be choosen to be complemented. (ii) By [8, Proposition 4.3], L∗ (s) is isomorphic as a topological ∗ -algebra to the matrix algebra Λ(A) := {x = (x ij ) ∈ CN : ∀N ∈ N ∃n ∈ N 2

∑ |x ij | max {

i,j∈N2

iN jN , } < ∞} jn in

endowed with the so-called Köthe-type PLB-space topology. Clearly, K∞ is a ∗ subalgebra of Λ(A). However, since K∞ is dense in Λ(A), this representation is not interesting for us. We have also a simillar phenomenon in the case of the algebra s. The diagonal matrices from Λ(A) give – also in the topological sense – exactly the space s󸀠 . In particular, the ∗ -algebra s has a simple representation in Λ(A). But s is dense in s󸀠 , so this representation does not work for us. In the proof of Theorem 4.1, Examples (vi) and (vii), we see that the representations of s and K∞ in L∗ (s) are much more sofisticated. Finally, we shall give an example of a unital commutative Fréchet ∗ -algebra isomorphic as a Fréchet space to s which is not a DN-algebra. Let A∞ (D) be the space of holomorphic functions on the open unit disc D which are smooth up to boundary. In other words, A∞ (D) := {f ∈ A(D) : f (k) ∈ A(D) for all k ∈ N}, where A(D) is the disc algebra. The space A∞ (D) admits a natural Fréchet space topology given by the norms ||f||p := sup{|f (j) (z)| : z ∈ D , 0 ≤ j ≤ p},

p ∈ N0

(18)

and it is isomorphic to the space s (cf. [19, Section 2]). Moreover, A∞ (D) becomes a unital commutative Fréchet ∗ -algebra when endowed with the usual multiplication of functions and involution f ∗ (z) := f(z). The algebra A∞ (D) is thus a (dense) ∗ -subalgebra of the disc algebra.

32 | Ciaś Proposition 4.3. The algebra A∞ (D) is not isomorphic to any closed ∗ -subalgebra of L∗ (s). Proof. By Theorem 3.5, it is enough to show that there is no DN-norm on A∞ (D). Let (|| ⋅ ||p )p∈N be the fundamental sequence of norms on the space A∞ (D) defined by (18). First, we will show that the norm || ⋅ ||[−1,1] , ||f||[−1,1] := sup |f(x)|, x∈[−1,1]

is not a dominating norm on A∞ (D). Let f n (z) := (z2 − 1)n for n ∈ N and z ∈ D. Let us fix n ∈ N and 0 ≤ k ≤ n. Then, clearly, ||f n ||[−1,1] = 1 and ||f n ||0 = 2n . Moreover, by the Leibniz rule, we obtain 󵄨󵄨 󵄨󵄨 k k d j d k−j (k) |f n (z)| = 󵄨󵄨󵄨󵄨 ∑ ( )( ) (z − 1)n ( ) (z + 1)n 󵄨󵄨󵄨󵄨 dz dz 󵄨 󵄨 j=0 j

󵄨󵄨 k k 󵄨󵄨 = 󵄨󵄨󵄨󵄨 ∑ ( ) ⋅ n(n − 1) . . . (n − j + 1)(z − 1)n−j ⋅ n(n − 1) . . . (n − k + j + 1)(z + 1)n−k+j 󵄨󵄨󵄨󵄨 󵄨 j=0 j 󵄨 ≤ n k 2k ⋅

sup

z∈D,0≤j≤k

= n k 2 2 2n ⋅ k

|z − 1|n−j |z + 1|n−k+j

sup 0≤θ≤ 2π ,0≤j≤k

for all z ∈ D, and thus

(1 − cos θ)

n−j 2

(1 + cos θ)

n−k+j 2

≤ n k 2k 2n

||f n ||p ≤ n p 2p 2n

for every p ∈ N0 . Consequently, for every p ∈ N and every C > 0 there is n ∈ N such that ||f n ||20 = 4n > Cn p 2p 2n ≥ C||f n ||[−1,1] ||f n ||p ,

and therefore || ⋅ ||[−1,1] is not a dominating norm on A∞ (D). Now, let || ⋅ || = √(⋅, ⋅) be an abitrary continuous norm on A∞ (D) satisying condition (α) in Definition 3.1. We define a continuous linear functional φ on A∞ (D) by φ(f) := (f, 1), where 1 is the identically one function. Then, by condition (α), φ(f ∗ f) = (f ∗ f, 1) = ||f||2 ≥ 0 for every f ∈ A∞ (D), so φ is a positive functional. In [16], there is an elementary proof of the fact that each positive linear functional on the disc algebra A(D) has some simple integral representation. It appears that the same proof works in the case of the algebra A∞ (D), and thus there is a positive Borel measure μ on [−1, 1] such that φ(f) = ∫ f(x)dμ(x) 1

−1

Fréchet algebras |

33

for every f ∈ A∞ (D). Hence, ||f||2 = ∫ |f(x)|2 dμ(x). 1

−1

If || ⋅ || was a dominating norm on A∞ (D) then, since || ⋅ || ≤ μ([−1, 1])|| ⋅ ||[−1,1] , the norm ||f||[−1,1] would be a dominating norm as well, contrary to our first claim. Hence, A∞ (D) is not a DN-algebra, which completes the proof. Acknowledgement: The author wishes to express his thanks to Leonhard Frerick for many stimulating conversations, especially during the stay at Trier University in June 2017. The author is also very endebted to his colleague Krzysztof Piszczek for valuable comments during the manuscript preparation.

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

[11] [12] [13]

T. Bloom, N. Levenberg, Pluripotential energy and large deviation. Indiana Univ. Math. J. 62 (2013), no. 2, 523–550. J. Bonet, P. Domański, The structure of spaces of quasianalytic functions of Roumieu type. Arch. Math. (Basel) 89 (2007), no. 5, 430–441. I. Chavel, Eigenvalues in Riemannian geometry. Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984. I. Chavel, Riemannian geometry. A modern introduction. Second edition. Cambridge Studies in Advanced Mathematics, 98. Cambridge University Press, Cambridge, 2006. T. Ciaś, On the algebra of smooth operators. Studia Math. 218 (2013), no. 2, 145–166. T. Ciaś, Commutative subalgebras of the algebra of smooth operators. Monatsh. Math. 179 (2016), no. 4, 1–23. T. Ciaś, Characterization of commutative algebras embedded into the algebra of smooth operators. Bull. London Math. Soc. 49 (2017), 102–116. T. Ciaś, K. Piszczek, The multiplier algebra of the noncommutative Schwartz space. Banach J. Math. Anal. 11 (2017), no. 3, 615–635. J. Dixmier, C ∗ -algebras. North-Holland Mathematical Library, Vol. 15. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. P. Domański, Classical PLS-spaces: spaces of distributions, real analytic functions and their relatives. Orlicz centenary volume, 51–70, Banach Center Publ., 64, Polish Acad. Sci. Inst. Math., Warsaw, 2004. P. Domański, D. Vogt, A splitting theory for the space of distributions. Studia Math. 140 (2000), no. 1, 57–77. L. Frerick, Extension operators for spaces of infinite differentiable Whitney jets. J. reine angew. Math. 602 (2007), 123–154. M. Măntoiu, R. Purice, On Fréchet-Hilbert algebras. Arch. Math. (Basel) 103 (2014), no. 2, 157–166.

34 | Ciaś

[14] R. Meise, D. Vogt, Introduction to functional analysis. Oxford University Press, New York 1997. [15] P. Michor, Answer avaible at http://mathoverflow.net/questions/173588/bases-for-spaces-of-smooth-functions? answertab=votes#tab-top. [16] M. Pavone, On the positive linear functionals on the disc algebra. Enseign. Math. (2) 35 (1989), no. 1-2, 51–54. [17] H.H. Schaefer, Topological vector spaces. Third printing corrected, Graduate Texts in Mathematics, Vol. 3, Springer-Verlag, New York-Berlin, 1971. [18] K. Schmüdgen, Unbounded Operator Algebras and Representation Theory. AkademieVerlag, Berlin, 1990. [19] B. A. Taylor, D. L. Williams, Ideals in rings of analytic functions with smooth boundary values. Canad. J. Math. 22 (1970), 1266–1283. [20] M. Tidten, Fortsetzungen von C ∞ -Funktionen, welche auf einer abgeschlossenen Menge in Rn definiert sind. Manuscr. Math. 27 (1979), no. 3, 291–312. [21] D. Vogt, Charakterisierung der Unterräume von s. Math. Z. 155 (1977), 109–117. [22] D. Vogt, M. J. Wagner, Charakterisierung der Quotienträume von s und eine Vermutung von Martineau. Studia Math. 67 (1980), 225–240. [23] D. Vogt, Unitary endomorphims of power series spaces. Math. Forum, Vol. 7, Studies on Mathematical Analysis, Vladikavkaz 2013, 220–239.

Antoine Derighetti

Relations between ideals of the Figà-Talamanca Herz algebra A p (G) of a locally compact group G and ideals of A p (H) of a closed subgroup Abstract: Let G be a locally compact group and H a closed subgroup. In analogy with the classical case, we obtain the two following results. Suppose at first that G is amenable and that I is a closed ideal of A p (H) having a bounded approximate unit, then the ideal {u ∈ A p (G) | ResH u ∈ I} of A p (G) also has a bounded approximate unit. The second result concerns the closedness of {ResH u | u ∈ I} in A p (H) for a closed ideal I of A p (G). We show that this set is closed if H is amenable. Keywords: locally compact group, amenability, convolution operator, pseudomeasure, Figà-Talamanca Herz algebra, ideal. Classification: Primary 43A15, 43A07 Secondary 43A22, 22D15.

1 Introduction Let G be a locally compact abelian group and H a closed subgroup, the Weil map T H relates the closed ideals of L1 (G) with the closed ideals of L1 (G/H) (see [11]). In the following, G denotes an arbitrary locally compact group (non necessarily commutative) and H a closed subgroup. We replace the Banach algebra L1 (G) of the commutative case, by the Eymard Fourier algebra denoted A2 (G) or more generally by the Figà-Talamanca Herz algebra A p (G) for 1 < p < ∞. For the definition of the Figà-Talamanca Herz algebra we refer to [5, Section 3.1, page 33-34]. The algebra L1 (G/H) is replaced by A p (H) and the Weil map is replaced by the restriction map ResH from A p (G) to A p (H) (see [5, Section 7.8]). Our first main result concerns closed ideals of A p (G). Let I be a closed ideal of A p (H). Assuming H amenable, we show (Theorem 3.3) that the closed ideal of A p (G) {u ∈ A p (G) | ResH u ∈ I} has a bounded approximate unit if the ideal I has a bounded approximate unit. Our proof gives a precise bound for the approximate unit of the ideal {u ∈ A p (G) | ResH u ∈ I}. For G non abelian the existence of the approximate unit cannot be obtained by elementary tools. Even for G abelian Antoine Derighetti, Ecole Polytechnique fédérale de Lausanne, EPFL SB-DO, MA A1 354 (Bâtiment MA), Station 8, CH-1015 Lausanne, Switzerland, e-mail: antoine.derighetti@epfl.ch https://doi.org/10.1515/9783110602418-0003

36 |

Derighetti

this existence is far to be trivial. To obtain our result we have to work with the multipliers of A p (G). Our second main result is also related to the Weil map T H . According to Hans Reiter the image under T H of a closed ideal of L1 (G) is a closed ideal of L1 (G/H) ([10] Lemma 1.1 page 553). We investigate whether for a closed ideal I of A p (G), the ideal {ResH u | u ∈ I} is closed in A p (H). We show (Corollary 4.6) that this ideal is closed if the subgroup H is amenable.

2 On the multipliers of A p (G) Let G be a locally compact group and 1 < p < ∞. A function u : G 󳨃→ C is said to be a multiplier of A p (G) if uv ∈ A p (G) for every v ∈ A p (G). The set of all multipliers of A p (G) is denoted MA p (G). This set is contained in the vector space C(G) of all continuous functions on G. For the pointwise product it is a subalgebra of C(G), involutive for the complex conjugation, with the unit 1G . Clearly MA p (G) contains A p (G). For every v ∈ MA p (G) there is C > 0 such that ||vu||A p (G) ≤ C||u||A p (G) for every u ∈ A p (G). This permits to put for v ∈ MA p (G) ||v||MA p (G) = sup{||vu||A p (G) | u ∈ A p (G) with ||u||A p (G) ≤ 1}. With this norm MA p (G) is an involutive unital normed algebra. This normed algebra is not complete in general. I mention two non trivial results on MA p (G): A p (G) ⊂ MA p (G d ) ([5] Section 3.3 Corollary 6 page 44), A2 (G) ⊂ MA p (G) ([5] Section 8.3 Theorem 8 page 158), with the corresponding inequalities for the norms. For T ∈ PM p (G) (for the definition of PM p (G) see [5] page 48) and for v ∈ MA p (G) the map L : u 󳨃→ ⟨uv, T⟩A p (G),PM p (G) belongs to A p (G)󸀠 . We have ||L||A p (G)󸀠 ≤ ||v||MA p (G) ||T||.

We put

v T = (Ψ G ) (L) p −1

p

(see [5, page 49] for the definition of Ψ G ). We denote by HomA p (G) PM p (G) the set of all norm continuous linear maps Φ of PM p (G) into itself such that Φ(uT) = u Φ(T)

Relations between ideals of A p (G) and ideals of A p (H) |

37

for every u ∈ A p (G) and every T ∈ PM p (G). For every Φ we put ||Φ|| = {||Φ(T)|| | T ∈ PM p (G), ||T|| ≤ 1}. For the composition of the maps, HomA p (G) PM p (G) is a Banach algebra. For v ∈ MA p (G) we put for every T ∈ PM p (G) λ(v)T = vT. Then λ is an isometric isomorphism of the normed algebra MA p (G) into HomA p (G) PM p (G). Denote by cv p (G) the norm closure in CV p (G) of {T ∈ CV p (G) | supp T is compact}. For every φ ∈ cv p (G)󸀠 and every T ∈ PM p (G) there a unique σ G (φ)(T) ∈ PM p (G) such that φ(uT) = < u, σ G (φ)(T) >

for every u ∈ A p (G). If G is amenable, σ G is a Banach space isometry of cv p (G)󸀠 onto HomA p (G) PM p (G), considered as a Banach space. We define a product on cv p (G)󸀠 in the following way: for φ ∈ cv p (G)󸀠 and T ∈ cv p (G) we put φ ⋅ T = σ G (φ)(T), for φ1 , φ2 ∈ cv p (G)󸀠 we set (φ1 ⋅ φ2 )(T) = φ1 (φ2 ⋅ T) for every T ∈ cv p (G). With this product, cv p (G)󸀠 is a Banach algebra. The map σ G is an isometric isomorphism of the Banach algebra cv p (G)󸀠 onto the Banach algebra HomA p (G) PM p (G).

3 Ideals with bounded approximate unit Let A be a commutative normed algebra. We say that A has a bounded approximate unit if there exists C ≥ 0 such that for every ε > 0 and for every a ∈ A there is b ∈ A with ||b|| ≤ C and ||a − ab|| < ε. The real number C is called a bound of the approximate unit. The infimum of all bounds of bounded approximate units is also a bound of approximate unit. This infimum is called the best bound of the bounded approximate units of A. To investigate ideals of A with bounded approximate unit we will use (see [2]) the structure of right A-module of the dual of A: for every f ∈ A󸀠 and for every a ∈ A we put (fa)(u) = f(au) for every u ∈ A. We denote by HomA A󸀠 the Banach algebra of all linear norm continuous maps P of A󸀠 into A󸀠 such that P(fa) = P(f)a for all f ∈ A󸀠 and all a ∈ A.

Proposition 3.1. Let A be a commutative normed algebra, C ≥ 0 and I a closed ideal of A. Suppose that A has an approximate unit bounded by one. (1) The following two statements are equivalent: (i) I has an approximate unit bounded by C, (ii) there is P ∈ HomA A󸀠 , a projection of A󸀠 onto I ⊥ , such that ||Id − P|| ≤ C.

38 |

Derighetti

(2) If I has a bounded approximate unit, there is P0 ∈ HomA A󸀠 , a projection of A󸀠 onto I ⊥ , such that ||Id − P0 || = inf{||Id − P|| | P ∈ HomA A󸀠 projection of A󸀠 onto I ⊥ }. This infimum is the best bound of the bounded approximate units of I. We denote by i the canonical inclusion of CV p (H) into CV p (G) ([5] Definition 7 page 106). Lemma 3.2. Let G be a locally compact group, 1 < p < ∞, H a closed subgroup of G and I a closed ideal of A p (H). Suppose that H is a set of p -synthesis in G. Then J ⊥ = i(I ⊥ ) where J = {u ∈ A p (G) | ResH u ∈ I}.

Proof. 1) J ⊥ ⊂ i(I ⊥ ). Let T ∈ J ⊥ . We claim that supp T ⊂ H. Otherwise there is x0 ∈ supp T with x0 ∈ ̸ H. There is U open neighborhood of x0 in G with U ∩ H = 0. By Theorem 1 page 88 of [5] there is u ∈ A p (G) with supp u ⊂ U and ⟨u, T⟩ ≠ 0. But from ResH u = 0 it follows u ∈ J and consequently ⟨u, T⟩ = 0. This is contradictory. According to Lohoué ([9]) there is S ∈ PM p (H) with i(S) = T. We show that S ∈ I⊥ . Let u ∈ I. There is v ∈ A p (G) with ResH v = u ([5] Theorem 5 page 140). We have v ∈ J and therefore ⟨u, S⟩ = ⟨ResH v, S⟩ = ⟨v, i(S)⟩ = 0.

This implies S ∈ I ⊥ . 2) i(I ⊥ ) ⊂ J ⊥ . Let T = i(S) with S ∈ I ⊥ . For v ∈ J we have ⟨v, T⟩ = ⟨ResH v, S⟩ but ResH v ∈ I consequently ⟨ResH v, S⟩ = 0 and therefore T ∈ J ⊥ .

Let G be a locally compact group and H a closed subgroup of G. We say that G ∈ [A]H if for every ε > 0, for every compact subset K of H and for neighborhood U of e in G there is f ∈ L 1 (G/H, m G/H ) such that N1 (f) = 1, f ≥ 0, supp f ⊂ ω(U) (ω being the canonical map of G onto G/H) and −1 ̇ ̇ ̇ ẋ < ε x) − f(x)|d ∫ |χ(h−1 , x)f(h

G/H

for every h ∈ K. In [4] we proved for G ∈ [A]H and 1 < p < ∞ the existence of a linear norm continuous map P of CV p (G) into CV p (H) such that (1) ||P(T)|| ≤ ||T|| for every T ∈ CV p (G), ( 2) P(i(S)) = S for every S ∈ CV p (H),

Relations between ideals of A p (G) and ideals of A p (H)

( 3) ( 4) ( 5) ( 6)

| 39

P(i(S) T i(S󸀠 )) = S P(T) S󸀠 for every S, S󸀠 ∈ CV p (H) and for every T ∈ CV p (G), P(uT) = ResH (u)P(T) for every u ∈ A p (G) and every T ∈ CV p (G), supp P(T) ⊂ supp T for every T ∈ CV p (G), p p P(λ G (μ)) = λ H (ResH μ) for every bounded measure μ.

Theorem 3.3. Let G be a locally compact amenable group, H an arbitrary closed subgroup of G, 1 < p < ∞, C ≥ 0 and I a closed ideal of A p (H) having an approximate unit bounded by C. Then the ideal {u ∈ A p (G) | ResH u ∈ I} has an approximate unit bounded by C + 2.

Proof. The group G being amenable, we have G ∈ [A]H . There is consequently a linear norm continuous map P of CV p (G) onto CV p (H) with the six properties described above. For every Φ ∈ HomA p (H) PM p (H) we put λ(Φ) = σ G ∘t P ∘ σ−1 H ∘ Φ. Then λ is a Banach algebra isometry of HomA p (H) PM p (H) into HomA p (G) PM p (G) and λ(Φ) = i ∘ Φ ∘ P for every Φ ∈ HomA p (H) PM p (H). According to Proposition 3.1 there is P ∈ HomA p (H) PM p (H) projection of PM p (H) onto I ⊥ with ||Id PM p (H) − P|| ≤ C.

1) The element of HomA p (G) PM p (G) λ(P) is a projection of PM p (G) onto J ⊥ where J = {u ∈ A p (G) | ResH u ∈ I}. We at first show that λ(P)(T) ∈ J ⊥ for every T ∈ PM p (G). Let v be an element of J. We have ⟨v, λ(P)T⟩ = ⟨v, i(P(P(T)))⟩ = ⟨ResH v, P(P(T))⟩.

From ResH v ∈ I and P(P(T)) ∈ I ⊥ it follows that

and consequently

⟨ResH v, P(P(T))⟩ = 0 ⟨v, λ(P)T⟩ = 0.

To finish the proof of 1) it suffices to verify that λ(P)T = T for every T ∈ J ⊥ . According to Lemma 3.2 there is S ∈ I ⊥ such that i(S) = T. Consider an arbitrary u ∈ A p (G). We have ⟨u, λ(P)T⟩ = ⟨ResH u, P(P(T))⟩ = ⟨ResH u, P(P(i(S)))⟩

But P(S) = S. We finally get

= ⟨ResH u, P(S)⟩.

⟨u, λ(P)T⟩ = ⟨ResH u, S⟩ = ⟨u, i(S)⟩ = ⟨u, T⟩.

40 |

Derighetti

2) The ideal J has an approximate unit bounded by C + 2. It suffices (see Proposition 3.1) to verify that ||Id PM p (G) − λ(P)|| ≤ C + 2. We have

||Id PM p (G) − λ(P)|| ≤ ||Id PM p (G) − i ∘ P|| + ||i ∘ P − λ(P)|| ≤ 2 + ||i ∘ P − λ(P)||.

Now for every T ∈ PM p (G) with ||T|| ≤ 1 we have

||i(P(T)) − i(P(P(T)))|| = ||P(T) − P(P(T))|| = ||(Id PM p (H) − P)(P(T))|| ≤ ||Id PM p (H) − P|| ||P(T)|| ≤ C. This implies

||i ∘ P − λ(P)|| ≤ C

and finally

||Id PM p (G) − λ(P)|| ≤ C + 2.

Proposition 3.4. Let G be a locally compact amenable group, H a closed subgroup of G and 1 < p < ∞. We suppose that H is nonopen in G. The following two statements are verified. (1) There is P ∈ HomA p (G) (PM p (G)) which is a projection of PM p (G) onto {u ∈ A p (G) | ResH u = 0}⊥ . (2) For every P ∈ HomA p (G) (PM p (G)), projection of PM p (G) onto {u ∈ A p (G) | ResH u = 0}⊥ , we have P(T) = 0 for every T ∈ PF p (G).

Proof. We put J = {u ∈ A p (G) | ResH u = 0}. The statement (1) is a consequence of Theorem 3.3 and Proposition 3.1. To prove (2) we consider f ∈ C00 (G) and u ∈ A p (G). Let ε be an arbitrary positive real number. We set K = suppf ∗ . The subgroup H being locally m G -negligible in G, there is an open neighborhood U of K ∩ H in G such that m G (U)
0, there is v ∈ A2 (H) such that ||u − uv||A2 (H) < ε. Then the ideal ResH I is closed in A2 (H). Proof. According to Dixmier PM2 (G) = CV2 (G) and therefore H is a set of 2synthesis in G ([5] Lemma 3 page 143). Remark 4.4. For SO(1, n), SU(1, n) and Sp(1, n) and their closed subgroups, A2 (G) verify the condition (2). Corollary 4.5. Let G be a locally compact group, H a closed subgroup of G, 1 < p < ∞ and I a closed ideal of A p (G). Suppose that the following three conditions are verified: (1) the subgroup H is locally neutral in G, (2) G/H admits a G -invariant measure, (3) the Banach algebra A p (H) has an approximate identity in the following weak sense: for every u ∈ A p (H) for every ε > 0 there is v ∈ A p (H) such that ||u − uv||A p (H) < ε.

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Then the ideal ResH I is closed in A p (H). Proof. According to Proposition 8 page 242 of [4] we have G ∈ [A]H . The set H is of p-synthesis in G by the Corollary 4 of [3] Corollary 4.6. Let G be a locally compact group, H a closed amenable subgroup of G, 1 < p < ∞ and I a closed ideal of A p (G). Then the ideal ResH I is closed in A p (H).

Proof. By [4] G ∈ [A]H . By Carl Herz H is a set of p -synthesis in G ([5] Section 7.9 Theorem 5 p. 144). Remark 4.7. In [1] we proved this result for H closed normal amenable subgroup of G.

Bibliography [1]

J. Delaporte and A. Derighetti,p -Pseudomeasures and Closed Subgroups, Monatsh. Math. 119, 37-47 (1995). [2] J. Delaporte and A. Derighetti,Best bounds for the approximate units of certain ideals of L1 (G) and of A p (G), Proc. Am. Math. Soc. 124 (1996) 1159-1169. [3] J. Delaporte and A. Derighetti,Invariant projections and convolution operators, Proc. Am. Math. Soc. 129 (2001) 1427-1435 . [4] A. Derighetti, Conditional expectations on CV p (G). Applications, J. Funct. Anal. 247 (2007), 231-251. [5] A. Derighetti,Convolution Operators on Groups, Lecture Notes of the Unione Matematica Italiana, Springer-Verlag Berlin Heidelberg 2011. [6] A. Derighetti, Sets of p -uniqueness on noncommutative locally compact groups, Colloq. Math. 149 (2017) n∘ 2, 257-263. [7] B. Forrest, E. Kaniuth, A.T. Lau and N. Spronk,Ideals with bounded approximate identities in the Fourier algebra, J. Funct. Anal., 203 (2003), 286-304. [8] B. Forrest and N. Spronk,Best bounds for approximate identities in ideals of the Fourier algebra vanishing on subgroups, Proc. Am. Math. Soc. 134 (2005), 111-116. [9] N. Lohoué, Estimations L p des coefficients de représentation et opérateurs de convolution, Adv. Math. 38 (1980), 178-221. [10] H. Reiter, Contributions to harmonic analysis: VI, Ann. of Math. 77, 552-562 (1963). [11] H. Reiter and J. D. Stegeman,Classical Harmonic Analysis and Locally Compact Groups, Clarendon Press. Oxford 2000.

Y. Estaremi

The composition of conditional expectation and multiplication operators Abstract: In this paper, we give some necessary and sufficient conditions for weighted conditional type operators on L2 (Σ) to be centered. Consequently, under a weak condition we get that the weighted conditional type operator M w EM u is centered if and only if the conditional type Hölder inequality turn into equality for w, u. As an important consequence we get that the Aluthge transformations of a conditional type operator is always centered and it’s point spectrum and joint point spectrum are equal. Also, we obtain that normality, hyponormality, p-hyponormality and centeredness of all operators of the form of EM u are coincided. In addition we get that all operators of the form M v M ū EM u are always normal and so centered, when the function v is A-measurable. Moreover, we see that the point spectrum and joint-point spectrum of weighted conditional type operators are equal, when they are centered. As an important consequence, we get that the point spectrum and joint point spectrum of all operators of the form M v M ū EM u on L2 (Σ) are equal when v is A-measurable and S(vE(u|2 )) = X. Keywords: Conditional expectation, normal operator, centered operator, point spectrum, joint point spectrum. Classification: 47B20, 47B38.

1 Introduction and Preliminaries Operators in function spaces defined by conditional expectations were first studied, among others, by S - T.C. Moy [16], Z. Sidak [19] and H.D. Brunk [4] in the setting of L p spaces. Conditional expectation operators on various function spaces exhibit a number of remarkable properties related to the underlying structure of the given function space or to the metric structure when the function space is equipped with a norm. P.G. Dodds, C.B. Huijsmans and B. de Pagter [5] linked these operators to averaging operators defined on abstract spaces earlier by J.L. Kelley [13], while A. Lambert [15] studied their link to classes of multiplication operators which form Hilbert C∗ -modules. J.J. Grobler and B. de Pagter [10] showed that the classes of partial integral operators, studied by A.S. Kalitvin and othY. Estaremi , Department of Mathematics, Payame Noor University, p. o. box: 19395-3697, Tehran, Iran, email: [email protected] https://doi.org/10.1515/9783110602418-0004

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ers [1, 2, 3, 6, 12], were a special case of conditional expectation operators. Recently, J. Herron studied operators EM u on L p spaces in [11]. Theory of conditional type operators is one of important arguments in the connection of operator theory and measure theory. The class of weighted conditional type operators contains composition operators, multiplication operators, weighted composition operators, some integral type operators and etc. These are some reasons that stimulate us to consider weighted conditional type operators in our work. Let f ∈ L0 (Σ), then f is said to be conditionable with respect to E if f ∈ D(E) := {g ∈ L0 (Σ) : E(|g|) ∈ L0 (A)}.

Throughout this paper we take u and w in D(E). Let (X, Σ, μ) be a complete σ-finite measure space. For any sub-σ-finite algebra A ⊆ Σ, the L2 -space L2 (X, A, μ|A ) is abbreviated by L2 (A), and its norm is denoted by ||.||2 . All comparisons between two functions or two sets are to be interpreted as holding up to a μ-null set. The support of a measurable function f is defined as S(f) = {x ∈ X; f(x) ≠ 0}. We denote the vector space of all (equivalence classes of) almost everywhere finite valued measurable functions on X by L0 (Σ). For a sub-σ-finite algebra A ⊆ Σ, the conditional expectation operator associated with A is the mapping f → EA f , defined for all non-negative, measurable function f as well as for all f ∈ L2 (Σ), where EA f , by the Radon-Nikodym theorem, is the unique A-measurable function satisfying ∫ fdμ = ∫ EA fdμ, (A ∈ A). A

A

As an operator on is idempotent and EA (L2 (Σ)) = L2 (A). If there is no possibility of confusion, we write E(f) in place of EA (f). A detailed discussion of the properties of conditional expectation may be found in [18]. Let u ∈ L0 (Σ). Then u is said to be conditionable with respect to E if L2 (Σ),

EA

u ∈ D(E) := {f ∈ L0 (Σ) : E(|f|) ∈ L0 (A)}. We now define the class of operator under investigation. Definition 1.1. Let (X, Σ, μ) be a σ-finite measure space and let A be a σ-subalgebra of Σ such that (X, A, A) is σ-finite. Let E be the corresponding conditional expectation operator on L2 (Σ) relative to A. If w, u ∈ L0 (Σ) such that uf is conditionable and wE(uf) ∈: L2 (Σ) for all f ∈ L2 (Σ), then the corresponding weighted conditional type operator is the linear transformation M w EM u : L2 (Σ) → L2 (Σ) defined by f → wE(uf).

We recall that an A-atom of the measure μ is an element A ∈ A with μ(A) > 0 such that for each F ∈ A, if F ⊆ A, then either μ(F) = 0 or μ(F) = μ(A). A measure space

Weighted conditional type operators

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49

(X, Σ, μ) with no atoms is called a non-atomic measure space. It is well-known fact that every σ-finite measure space (X, A, μ|A ) can be partitioned uniquely as X = (⋃n∈N A n ) ∪ B, where {A n }n∈N is a countable collection of pairwise disjoint A-atoms and B, being disjoint from each A n , is non-atomic (see [20]). An operator A on a Hilbert space is called centered, if the family of operators k n {A∗ A n , A k A∗ : n, k ≥ 0} is commutative [17]. In [7, 8, 9] and some other papers we investigated some classic properties of weighted conditional type operator M w EM u on L p -spaces. In this paper we will be concerned with characterizing weighted conditional expectation type operators and their Aluthage transformations on L2 (Σ) in terms of membership in the class of centered operators and the relation between normal and centered weighted conditional type operators. Moreover, we get some necessary and sufficient conditions for integral operators to be centered and some applications in their spectra. The results of [5] state that our results are valid for a large class of linear operators, since for finite measure space (X, Σ, μ) we have L∞ (Σ) ⊆ L2 (Σ) ⊆ L1 (Σ) and L2 (Σ) is an order ideal of measurable functions on (X, Σ, μ).

2 Centered and normal weighted conditional type operators In this section, first we recall some auxiliary results. Theorem 2.1. [8] The operator T = M w EM u is bounded on L2 (Σ) if and only if 1 1 (E|w|2 ) 2 (E|u|2 ) 2 ∈ L∞ (A), and in this case its norm is given by ||T|| = ||(E(|w|2 )) 2 (E(|u|2 )) 2 ||∞ . 1

1

Lemma 2.2. [8] Let T = M w EM u be a bounded operator on L2 (Σ) and let p ∈ (0, ∞). Then 2 ))p−1 χ (E(|w|2 ))p EM u (T ∗ T)p = M u(E(|u| ̄ S and

(TT ∗ )p = M w(E(|w|2 ))p−1 χ G (E(|u|2 ))p EM w̄ ,

where S = S(E(|u|2 )) and G = S(E(|w|2 )).

Theorem 2.3. [8] The unique polar decomposition of the bounded operator T = M w EM u is U|T|, where E(|w|2 ) 2 ̄ |T|(f) = ( ) χ S uE(uf) E(|u|2 ) 1

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and

2 χ S∩G U(f) = ( ) wE(uf), E(|w|2 )E(|u|2 ) 1

for all f ∈ L2 (Σ).

It is shown in [17] that if A is an operator such that for each positive n, A n has polar decomposition U n P n , then A is centered if and only if for each positive n, U n = U1n . In the sequel we will characterize centered weighted conditional type operators. Theorem 2.4. Consider the following weighted conditional type operator M w EM u : L2 (Σ) → L2 (Σ). (a) If M w EM u is centered, then |E(uw)|2 = E(|u|2 )E(|w|2 ) on S(E(uw)). (b) If |E(uw)|2 = E(|u|2 )E(|w|2 ), then M w EM u is centered.

Proof. (a) If T = M w EM u is such that T ∗ T and TT ∗ commute (in particular if T centered), then the identity M

1

(E(|w|2 )) 2

EM u [T ∗ T, TT ∗ ]M w EM

1

(E(|w|2 )) 2

=0

quickly leads to |E(uw)|2 = E(|u|2 )E(|w|2 ) on S(E(uw)). (b) Suppose that |E(uw)|2 = E(|u|2 )E(|w|2 ). By the method of part (a) and direct computation shows that U n = U1n . Thus M w EM u is centered. Here as a corollary of Theorem 2.4 we get that under a weak condition the operator M w EM u is centered if and only if the conditional type Hölder inequality is an equality for w, u. Corollary 2.5. If S(E(uw)) = X, then the operator M w EM u on L2 (Σ) is centered if and only if |E(uw)|2 = E(|u|2 )E(|w|2 ).

If we set w = 1 in Theorem 2.4, then for the operator EM u we have the following remark. Remark 2.6. Consider the following weighted conditional type operator EM u : L2 (Σ) → L2 (Σ). (a) If EM u is centered, then |E(u)|2 = E(|u|2 ) on S(E(u)). (b) If |E(u)|2 = E(|u|2 ) or equivalently u ∈ L0 (A), then EM u is centered.

By using some properties of the conditional expectation operator E we have the following fundamental lemma. Lemma 2.7. If u ≥ 0, then S(E(u)) = S(E(|u|2 )).

Proof. Suppose that u ≥ 0. So we have

S(u) ⊆ S(E(u)) = S((E(u))2 ) ⊆ S(E(u2 )).

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Since S(u) = S(u2 ) ⊆ S(E(u2 )) and S(E(u)) is the smallest A-measurable set so that contains S(u), then we get that S(E(u)) = S(E(|u|2 )). So the Lemma 2.7 gives us the next remark. Remark 2.8. (a) If S(E(u)) = S(E(|u|2 )), then the operator EM u on L2 (Σ) is centered if and only if u ∈ L0 (A). (b) If u ≥ 0, then EM u is centered if and only if u ∈ L0 (A).

Recall that each operator A on a Hilbert space H is called normal if A∗ A = AA∗ . In the sequel some necessary and sufficient conditions for normality will be presented.

Theorem 2.9. Let T = M w EM u be a bounded operator on L2 (Σ). 1 1 (a) If (E(|u|2 )) 2 w̄ = u(E(|w|2 )) 2 , then T is normal. (b) If T is normal, then |E(u)|2 E(|w|2 ) = |E(w)|2 E(|u|2 ). Proof. (a) Applying Lemma 2.2 we have 2 ) EM u − M wE(|u|2 ) EM w̄ . T ∗ T − TT ∗ = M uE(|w| ̄

So for every f ∈ L2 (Σ),

̄ f ̄ dμ ⟨T ∗ T − TT ∗ (f), f⟩ = ∫ E(|w|2 )E(uf)uf̄ − E(|u|2 )E(wf)w X

̄ 2 dμ. = ∫ |E(u(E(|w|2 )) 2 f)|2 − |E((E(|u|2 )) 2 wf)| 1

1

X

This implies that if (E(|u|2 )) 2 w̄ = u(E(|w|2 )) 2 , 1

1

then for all f ∈ L2 (Σ), ⟨T ∗ T − TT ∗ (f), f⟩ = 0, thus T ∗ T = TT ∗ . (b) Suppose that T is normal. By (a), for all f ∈ L2 (Σ) we have ̄ 2 dμ = 0. ∫ |E(u(E(|w|2 )) 2 f)|2 − |E((E(|u|2 )) 2 wf)| 1

1

X

Let A ∈ A, with 0 < μ(A) < ∞. By replacing f to χ A , we have

̄ 2 dμ = 0 ∫ |E(u(E(|w|2 )) 2 )|2 − |E((E(|u|2 )) 2 w)| 1

1

A

and so

∫ |E(u)|2 E(|w|2 ) − |E(w)|2 E(|u|2 )dμ = 0. A

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Since A ∈ A is arbitrary, then |E(u)|2 E(|w|2 ) = |E(w)|2 E(|u|2 ). Now we recall the form of the Aluthge transformation of a weighted conditional type operators. Theorem 2.10. [8] The Aluthge transformation of T = M w EM u is ̂ = χ S E(uw) uE(uf), ̄ T(f) E(|u|2 )

f ∈ L2 (Σ).

Hence we have the following proposition. Proposition 2.11. The Aluthge transformation of T = M w EM u is always centered.

̂ = M χS E(uw) EM u = T1 T2 , where T1 = M χS E(uw) and T2 = M ū EM u Proof. Since T ū

̂ is always such that T1 is a normal operator and T2 is a positive operator. Then T normal. And so it is always centered. E(|u|2 )

E(|u|2 )

In general all operators of the form M v M ū EM u such that v is an A-measurable function, are always normal and therefore are centered. In addition, it is shown in [7] that the operator M w EM u is normal if and only if is p-hyponormal. Also, in [8] it is shown that M w EM u is p-hyponormal if and only if is hyponormal. These observations give us the following remark. Remark 2.12. Consider the following weighted conditional type operator T = EM u : L2 (Σ) → L2 (Σ). Then the followings hold: (a) If S(E(u)) = S(E(|u|2 )), then the following conditions are mutually equivalent: (1) T is normal. (2) T is hyponormal. (3) T is p-hyponormal. (4) T is centered. (5) u ∈ L∞ (A). (b) If u ≥ 0, then the cases in part (a) are mutually equivalent.

Example 2.13. Let X = [−1, 1], dμ = 21 dx and A = ⟨{(−a, a) : 0 ≤ a ≤ 1}⟩ (σalgebra generated by symmetric intervals). Then EA (f)(x) =

f(x) + f(−x) , x ∈ X, 2

whenever EA (f) is defined. Let u(x) = e x , then E(u)(x) = cosh(x) and E(|u|2 )(x) = cosh(2x). It is clear that S(E(u)) = X. So, u is not A-measurable. Thus by Remark 2.9 we conclude that the operator T = EM u is not centered. If T = EM u is centered, then u should be A-measurable, but u is not A-measurable.

Weighted conditional type operators

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3 Applications In this section, we shall denote by σ p (T), σ jp (T), the point spectrum of T, the joint point spectrum of T, respectively. A complex number λ ∈ C is said to be in the point spectrum σ p (T) of the operator T, if there is a unit vector x satisfying (T − ̄ = 0, then λ is said to be in the joint spectrum λ)x = 0. If in addition, (T ∗ − λ)x σ jp (T) of T. If A, B ∈ B(H), then it is well known that σ p (AB) \ {0} = σ p (BA) \ {0} and σ jp (AB) \ {0} = σ jp (BA) \ {0}. In the sequel we assume that EM u and M w E are bounded operators on L2 (Σ). Here we recall a theorem of [7] that we will use in the next theorem. Theorem 3.1. [7] We consider T = M w EM u as an operator on L2 (Σ) and |E(uw)|2 = E(|u|2 )E(|w|2 ) a.e., μ. Then σ p (M w EM u ) = σ jp (M w EM u ). Now by using Theorem 3.1, Corollary 2.6 and Remark 2.7 we have the next theorem. Theorem 3.2. Let T = M w EM u be an operator on L2 (Σ). (a) If T is centered and S(E(uw)) = X, then σ p (M w EM u ) = σ jp (M w EM u ) and σ jp (M w EM u ) \ {0} = {λ ∈ C \ {0} : μ(A λ,w ) > 0},

where A λ,w = {x ∈ X : E(uw)(x) = λ}. (b) The point spectrum and joint point spectrum of all operators of the form M v M ū EM u on L2 (Σ) are equal when S(E(uw)) = X and σ p (M v ū EM u ) \ {0} = {λ ∈ C \ {0} : μ(A λ,v ū ) > 0}.

Proof. (i) Since T is centered and S(E(uw)) = X, then by Corollary 2.6 we have |E(uw)|2 = E(|u|2 )E(|w|2 ) a.e. So by Theorem 3.1 we get that σ p (M w EM u ) = σ jp (M w EM u ). Let A λ = {x ∈ X : E(u)(x) = λ}, for λ ∈ C. Suppose that μ(A λ ) > 0. Since A is σ-finite, there exists an A-measurable subset B of A λ such that 0 < μ(B) < ∞, and f = χ B ∈ L p (A) ⊆ L p (Σ). Now EM u (f) − λf = E(u)χ B − λχ B = 0.

This implies that λ ∈ σ p (EM u ). If there exists f ∈ L p (Σ) such that fχ C ≠ 0 μ-a.e, for C ∈ Σ of positive measure and E(uf) = λf for 0 ≠ λ ∈ C, then f = E(uf) λ , which means that f is A-measurable. Therefore E(uf) = E(u)f = λf and (E(u) − λ)f = 0. This implies that C ⊆ A λ and so μ(A λ ) > 0. This observations show that σ p (EM u ) \ {0} = {λ ∈ C \ {0} : μ(A λ ) > 0},

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therefore σ p (M w EM u ) \ {0} = {λ ∈ C \ {0} : μ(A λ,w ) > 0},

where A λ,w = {x ∈ X : E(uw)(x) = λ}. Statement (ii) is easily derived from Theorem 3.1. Hence we have the following corollaries.

Corollary 3.3. If T = M w EM u and E(uw)) = X, then the point spectrum and joint point spectrum of the Aluthge transformation of T are equal. Corollary 3.4. If the operator EM u is centered and S(E(u)) = X, then we have σ p (EM u ) = σ jp (EM u ) and σ jp (EM u ) \ {0} = {λ ∈ C \ {0} : μ(A λ ) > 0}. Here we show that a large class of integral operators are of the form of weighted conditional type operators. This means that we investigated centeredness and normality of integral operators on L2 (Σ). Let (X1 , Σ1 , μ1 ) and (X2 , Σ2 , μ2 ) be two σ-finite measure spaces and X = X1 × X2 , Σ = Σ1 × Σ2 and μ = μ1 × μ2 . Put A = {A × X2 : A ∈ Σ1 }. Then A is a sub-σ-algebra of Σ. Then for all f in domain EA we have EA (f)(x1 , x2 ) = ∫ f(x1 , y)dμ2 (y) μ − a.e. X2

on X. Also, if (X, Σ, μ) is a finite measure space and k : X × X → C is a Σ ⊗ Σmeasurable function such that ∫ |k(., y)f(y)|dμ(y) ∈ L2 (Σ) X

for all f ∈ L2 (Σ). Then the operator T : L2 (Σ) → L2 (Σ) defined by Tf(x) = ∫ k(x, y)f(y)dμ,

f ∈ L2 (Σ),

X

is called kernel operator on L2 (Σ)). We show that T is a weighted conditional type operator.[10] Since L2 (Σ) × {1} ≅ L2 (Σ) and uf is a Σ ⊗ Σ-measurable function,

Weighted conditional type operators

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55

when f ∈ L2 (Σ). Then by taking u := k and f 󸀠 (x, y) = f(y), we get that EA (uf)(x, y) = EA (uf 󸀠 )(x, y)

= ∫ u(x, y)f 󸀠 (x, y)dμ(y) X

= ∫ u(x, y)f(y)dμ(y) X

= Tf(x).

Hence T = EM u , i.e, T is a weighted conditional type operator. This means all assertions of this paper are valid for a class of integral type operators.

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

J. Appell, E. V. Frovola, A. S. Kalitvin and P. P. Zabrejko, Partial intergal operators on C([a, b] × [c, d]), Integral Equations and Operator Theory 27 (1997), 125 - 140. J. Appell, A.S. Kalitvin, and M.Z. Nashed, On some partial integral equations arising in the mechanics of solids, Journal of Applied Mathematics and Mechanics 79 (1999), 703 - 713. J. Appell, A.S. Kalitvin, and P.P. Zabrejko, Partial integral operators in Orlicz spaces with mixed norm, Collo. Math. 78 (1998), 293 - 306. H. D. Brunk, On an extension of the concept conditional expectation, Proc. Amer. Math. Soc. 14 (1963), 298-304. P.G. Dodds, C.B. Huijsmans and B. De Pagter, Characterizations of conditional expectation-type operators, Pacific J. Math. 141 (1990), 55-77. R. G. Douglas, Contractive projections on an L1 space, Pacific J. Math. 15 (1965), 443-462. Y. Estaremi, On a Class of Operators With Normal Aluthge Transformations, Filomat. 29 (2015), 969–975. Y. Estaremi and M.R. Jabbarzadeh, Weighted lambert type operators on L p -spaces, Oper. Matrices 1 (2013), 101-116. Y. Estaremi and M.R. Jabbarzadeh, Weighted composition lambert type operators on L p spaces, Mediterranean Journal of Mathematics 11 (2014), 955-964. J.J. Grobler and B. de Pagter, Operators representable as multiplication conditional expectation operators, Journal of Operator Theory 48 (2002) 15 - 40. J. Herron, Weighted conditional expectation operators, Oper. Matrices 1 (2011), 107-118. A.S. Kalitvin and P.P. Zabrejko, On the theory of partial integral operators, J. Integral Equations and Appl. 3 (1991), 351 - 382. J.L. Kelley, Averaging operators on C∞ (X), Illinois J. Math. 2 (1958), 214 - 223. A. Lambert, Conditional expectation related characterizations of the commutant of an abelian W ∗ -algebra, Far East J. of Math. Sciences 2 (1994), 1-7. A. Lambert, A Hilbert C ∗ -module view of some spaces related to probabilistic conditional expectation, Questiones Mathematicae 22 (1999), 165 - 170. Shu-Teh Chen, Moy, Characterizations of conditional expectation as a transformation on function spaces, Pacific J. Math. 4 (1954), 47-63.

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[17] B. B. Morrel and P. S. Muhly, Centered operators, Studia Math. 51 (1974), 251–263. [18] M. M. Rao, Conditional measure and applications, Marcel Dekker, New York, 1993. [19] S. Sidak, On relations between strict sense and wide sense conditional expectation, Theory of Probability and Applications 2 (1957), 267 – 271. [20] A. C. Zaanen, Integration, 2nd ed., North-Holland, Amsterdam, 1967.

Jean Esterle

On the generation of Arveson weakly continuous semigroups Abstract: We consider one-parameter semigroups T = (T(t))t>0 of bounded operators on a Banach space X which are weakly continuous in the sense of Arveson. For such a semigroup T set ω T (t) = ‖T(t)‖ for t > 0, and denote by L1ωT the space +∞ of (classes of) measurable functions f satisfying ∫0 |f(t)|ω T (t)dt < +∞. The Pettis integral ∫0 f(t)T(t)dt defines for f ∈ L1ωT a bounded operator ϕT (f) on X. We define the Arveson ideal IT of the semigroup to be the closure in B(X) of ϕT (L1ωT ). Using a variant of a procedure introduced a long time ago by the author we construct a norm-decreasing embedding from IT into a Banach algebra JT , called the normalized Arveson ideal of T. The image of IT is dense in JT , JT has a sequential bounded approximate identity, lim supt→0+ ‖T(t)‖B(JT ) < +∞, and there is a natural isomorphism from the "quasimultiplier algebra" QM(IT ) onto the "quasimultiplier algebra" QM(JT ), where QM(IT ) and QM(JT ) are equipped with their usual families of bounded subsets. The generator AT of the semigroup T is defined as a quasimultiplier on IT , or, equĩ T )}χ∈I valently, on JT . Let Res ar (AT ) be the complement of the set {χ(A IT ̂T , where ̂ denotes the space of characters on IT and where χ̃ denotes the unique extension of χ ∈ ̂ IT as a character on QM(IT ). The quasimultiplier A − μI has an inverse belonging to JT for μ ∈ Res ar (AT ), which allows to consider this inverse as a "regular" quasimultiplier on the Arveson ideal IT . The usual resolvent formula holds in this context for Re(μ) > limt→+∞ log‖T(t)‖ . t + Set Π α := {z ∈ C | Re(z) > α}. We revisit the functional calculus associated to the generator AT by defining F(−AT ) ∈ JT by a Cauchy integral when F belongs to the Hardy space H 1 (Π α+ ) for some α < − limt→+∞ log‖T(t)| . We then define F(−AT ) t as a quasimultiplier on JT and IT when F belongs to the Smirnov class on Π α+ , and F(−AT ) is a regular quasimultiplier on JT and IT if F is bounded on Π α+ . If F(z) = e−zt for some t > 0, then F(−AT ) = T(t), and if F(z) = −z, we indeed have F(−AT ) = AT . +∞

Keywords: semigroup of bounded operators, Arveson pairs, Arveson spectrum, Pettis integral, infinitesimal generator, resolvent, Laplace transform, holomorphic functional calculus. Classification: Primary 47A16; Secondary 47D03, 46J40, 46H20. Jean Esterle, IMB, UMR 5251, Université de Bordeaux, 351, cours de la Libération, 33405 Talence, France, email: [email protected] https://doi.org/10.1515/9783110602418-0005

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1 Introduction We study here the generator of a semigroup T = (T(t))t>0 of bounded operators on a Banach space X which is weakly continuous with respect to a dual pair (X, X∗ ) satisfying the conditions introduced by Arveson in 1974 in his seminal paper [2] on group representations. Semigroups satisfying some weaker properties have been considered in 1953 by Feller, who studied semigroups T = (T(t))t>0 of bounded operators on a Banach space X such there exists a nonzero continuous linear form l on X for which the map t →< T(t)x, l > is measurable for every x ∈ X, and his work involves the subspace X∗ of the dual space X 󸀠 generated by the family (l ∘ T(t))t>0 . Feller’s Pettis integrals allow to construct operators taking values on some subspace X̃ of X∗ , and he shows that there exists a rather large closed subspace X1 of X̃ such that the map t → T(t)x is continuous on (0, +∞) for x ∈ X1 . Set ωT (t) = ‖T(t)‖ for t > 0, and denote by MωT the convolution algebra con+∞ sisting in those measures μ on (0, +∞) such that ∫0 ωT d|μ|(t) < +∞. Arveson’s conditions are more restrictive, but they allow using Pettis integrals to define a +∞ norm-decreasing homomorphism ϕT : μ → ∫0 T(t)dμ(t) from the convolution Banach algebra MωT into B(X), without enlarging the Banach space X. Denote by L1ωT the convolution algebra of (classes) of measurable functions +∞ on the half-line satisfying ∫0 |f(t)|‖T(t)‖dt < +∞, identified to a closed ideal of MωT , and let AT (resp. IT ) be the closure of ϕT (MωT ) (resp ϕT (L1ωT )) in B(X) with respect to the operator norm. The ideal IT of AT is called the Arveson ideal associated to the semigroup T, see sections 2 and 3. In section 4 we introduce the infinitesimal generator AT,op as a weakly densely defined operator on the space X1 := [∪t>0 T(t)(X)]− , following Feller’s approach in [17], and we also introduce the resolvent R op (T, λ) = (AT,op − λI)−1 as a weakly densely defined operator on X1 . If lim supt→0+ ‖T(t)‖ < +∞, these partially defined operators are closed operators on X1 . In the general case they can be interpreted as closed operators on X0 := X1 / [∩t>0 Ker(T(t)] , which of course equals X1 if ∩t>0 Ker(T(t)) = {0}. In section 5 we pave the way to a more intrinsic approach to the infinitesimal generator of an Arveson weakly continuous semigroup T. It follows from some results of [11] that L1ωT contains a function g such that g ∗ L1ωT is dense in L1ωT , and so IT possesses dense principal ideals, and it follows from the weak continuity of the semigroup that uAT ≠ {0} for every u ∈ AT \ {0}. This allows us to apply to IT the theory of quasimultipliers developped a long time ago by the author in [12]. Recall that if U is a Banach algebra such that uU ≠ {0} for u ∈ U \ {0} and such that the set Ω(U) := {u ∈ U | [uU]− = U} is nonempty, a quasimultiplier on U is a closed densely defined operator on U of the form S = S u/v , where u ∈ U, v ∈

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Ω(U), and where the domain DS of S is the set of all x ∈ U such that ux ∈ vU. If x ∈ DS , then Sx is the unique y ∈ U such that ux = vy. A set ∆ ⊂ QM(U) is said to be pseudobounded if ∩S∈∆ DS contains some u ∈ Ω(U) such that supS∈∆ ‖Su‖ < +∞. A quasimultiplier S ∈ QM(U) is said to be regular if there exists λ > 0 such that the set {λ n S n }n≥1 is pseudobounded, and the family QM r (U) of all regular quasimultipliers on U is a pseudo-Banach algebra in the sense of [1]. In section 6, we associate in a natural way to a quasimultiplier S on IT a weakly densely defined operator S̃ on X1 . If ∩t>0 Ker(T(t)) = {0} then S̃ has an extension S̃ to a subspace of X1 which is a closed operator. In the general case the domain of S̃ can be considered as a subspace of X0 := X1 / ∩t>0 Ker(T(t)) and S̃ has an extension S̃ to a subspace of X0 which is a closed operator. In section 7 we use a variant of a construction introduced in [12] to define the "normalized Arveson ideal" JT . Set ρT := limt→+∞ ‖T(t)‖1/t . If λ > log(ρT ), then the family {e−λt T(t)}t>0 } is pseudobounded. In this case set, for u ∈ IT , ‖u‖λ = supt>0 ‖e−λt T(t)u‖ ∈ [0, +∞], set LT := {u ∈ IT | ‖u‖λ < +∞}, and denote by UT the closure of ∪t>0 T(t)IT in (LT , ‖.‖λ ). Then (LT , ‖.‖)λ is a Banach algebra, and the definition of LT and UT and the norm topology on LT and UT do not depend on the choice of λ > log(ρT ). We can identify the usual multiplier algebra M(UT ), which is a Banach algebra with respect to the operator norm ‖.‖op,λ , to a subalgebra of QMr (IT ). The closure JT of IT in (M(UT ), ‖.‖op,λ ) is called the normalized Arveson ideal of the semigroup T. The sequence (ϕT (f n ∗ δ ϵ n ))n≥1 is a sequential bounded approximate identity for JT for every Dirac sequence (f n )n≥1 and every sequence (ϵ n )n≥1 of positive real numbers which converges to 0, and the map S → S|UT is an isometric isomorphism from (M(JT ), ‖.‖M(JT ) ) onto (M(UT ), ‖.‖λ,op ). The algebras UT , IT and JT are similar in the sense of [12], and an explicit pseudobounded isomorphism between the quasimultiplier algebras QM(JT ) and QM(UT ) or QM(IT ) (resp. between the algebras QMr (JT ) and QMr (UT ) or QMr (IT ) ) is given by the map S u/v → S au/av where a ∈ Ω(UT ). We have Ω(JT ) ∩ UT ⊂ Ω(UT ), and the fact that Ω(JT ) ∩ UT ≠ 0 follows from a construction based on inverse Laplace transforms of outer functions on half-planes given by P. Koosis and the author in section 6 of [11]. In section 8 we consider the generator AT of the semigroup as a quasimultiplier on the algebra IT or JT . To define the infinitesimal generator as a quasimultiplier on IT we use the formula AT,IT = S−ϕT (f 󸀠 )/ϕT (f0 ) ,

where f0 ∈ C 1 ([0, +∞)) ∩ Ω (L1ωT ) satisfies f0 (0) = 0, f0󸀠 ∈ L1ωT . This definition does not depend on the choice of f0 and the quasimultiplier AT,IT is a closed operator on the Arveson ideal IT . If u ∈ IT , and if we have − v‖ = 0 for some v ∈ IT , then u belongs to the domain of limt→0+ ‖ T(t)u−u t AT,IT and AT,IT u = v. It is also possible to consider the infinitesimal generator 0

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of the semigroup as a quasimultiplier on the normalized Arveson ideal JT . Set then ω̃ T (t) = ‖T(t)‖M(JT ) , and define ϕ̃ T = Mω̃ T → M(JT ) by using the formula +∞

ϕ̃ T (μ) := ∫ T(t)dμ(t), 0

where the Bochner integral is computed with respect to the strong operator topology on M(JT ). To define the infinitesimal generator as a quasimultiplier on JT we use the formula AT,JT = S−ϕ̃ T (f 󸀠 )/ϕ̃ T (f0 ) ,

Ω (L1ω̃ T )

where f0 ∈ ∩ satisfies f0 (0) = 0, f0󸀠 ∈ L1ω̃ T . Notice that while the fact that C 1 ([0, +∞)) ∩ Ω (L1ωT ) ≠ 0 follows from nontrivial results from section 6 of [11] it is very easy to find elements f0 ∈ C 1 ([0, +∞)) ∩ Ω (L1ω̃ T ) satisfying f0 (0) = 0, f0󸀠 ∈ L1ωT : one can take for example f0 = v λ , for some λ > log(ρT ), where v λ (t) = te−λt for t ≥ 0. This definition agrees with the classical definition of the generator of a strongly continous semigroup bounded in norm near 0 : if u ∈ JT , then u belongs to the domain of AT,JT if − v‖JT = 0, and in this and only if there exists v ∈ JT such that limt→0+ ‖ T(t)u−u t situation AT,JT u = v. Identify the quasimultiplier algebras QM(IT ) and QM(JT ), and set AT = AT,IT = AT,JT . Denote by ̂ IT the space of characters on IT , equipped with the usual Gelfand topology, i.e. the topology induced on ̂ IT by the w∗ -topology on the unit ball of the dual of ̂ IT . Every character χ on IT extends in a unique way to ̃ T) a character χ̃ on QM(IT ), and there exists a unique complex number a χ = χ(A such that χ(T(t)) = e ta χ for t > 0. If the Arveson ideal IT is not a radical al̃ gebra, then the family σ ar (S) := {χ(S)} ̂T will be called the Arveson spectrum χ∈I of a quasimultiplier S ∈ QM(IT ), and we will use the convention σ ar (AT ) = 0 if ̃ T ) defines a homeothe algebra IT is radical. If IT is not radical, the map χ → χ(A morphism from ̂ IT onto σ ar (AT ), and we have, for f ∈ MωT , C 1 ([0, +∞))

0

̃ T (f)) = L(f)(−χ(A ̃ T )). χ(ϕ

̃ T )) for f ∈ Mω̃ T , see section 9. Similarly, we have χ(̃ ϕ̃ T (f)) = L(f)(−χ(A In section 10 we introduce the Arveson resolvent set Res ar (AT ) := {C \ σ ar (AT )}, and observe that AT − μI has an inverse (AT − μI)−1 in QM(IT ) for μ ∈ Res ar (AT ). More precisely this quasimultiplier (AT − μI)−1 belongs to JT ⊂ QMr (IT ), and the map μ → (AT − μI)−1 is an holomorphic map from Res ar (AT ) into JT . If Re(λ) > −log(ρT ), then we have the usual resolvent formula +∞

(A − λI)−1 = − ∫ e−λt T(t)dt ∈ JT , 0

where the Bochner integral is computed with respect to the strong operator topology on M(JT ).

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We can also define in this case (A − λI)−1 by using the formula +∞

(A − λI)−1 v = − ∫ e−λt T(t)vdt (v ∈ UT ), 0

which defines a quasimultiplier on IT if we apply it with v ∈ UT ∩ Ω(IT ). In section 11 we introduce a holomorphic functional calculus. For α ∈ R , set Π α+ := {z ∈ C | Re(z) > α}. Let H 1 (Π α+ ) be the usual Hardy space consisting in those functions F holomorphic on Π α+ satisfying ‖F‖1 := sup β>α

+∞

1 ∫ |F(β + iy)|dy < +∞. 2π −∞

Functions F ∈ H 1 (Π α+ ) admit a.e. a nontangential limit F ∗ (α + iy) on α + iR , and +∞ 1 ‖F‖1 := 2π ∫−∞ |F ∗ (α + iy)|dy. The restriction to Π β+ of a function F ∈ H 1 (Π α+ ) is bounded on Π β+ for β > α, and so ∪α 0. We will often write T = (T(t))t>0 . A semigroup will be said to be normalized if ∪t>0 T(t)(X) is dense in X with respect to the norm topology. We now introduce the following definition, which is an obvious extension to one-parameter semigroups of the notion of weakly continuous group representations associated to a dual pairing, due to Arveson [2]. Definition 2.3. Let (X, X∗ ) be a dual pairing, and let Bw (X) the space of weakly continuous elements of B(X). A semigroup T = (T(t))t>0 ⊂ Bw (X) is said to be weakly continuous with respect to the dual pairing (X, X∗ ) if the function given by t 󳨃→< T(t)x, l > is continuous on (0, +∞) for every x ∈ X and every l ∈ X∗ .

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Let (X, X∗ ) be a dual pairing, and let T = (T(t))t>0 ⊂ Bw (X) be a weakly continuous semigroup. We have, for t > 0, ‖T(t)‖ = sup{| < T(t)x, l > | : x ∈ X, ‖x‖ ≤ 1, l ∈ X∗ , ‖l‖ ≤ 1}

and so the weight ωT : t 󳨃→ ‖T(t)‖ is lower semicontinuous on (0, +∞). Let K ⊂ (0, +∞) be compact. Since supt∈K ‖T(t)x‖ < +∞ for every x ∈ X, it follows again from the Banach-Steinhaus theorem that supt∈K ‖T(t)‖ < +∞. A standard argument shows then that lim ‖T(t)‖ t = lim ‖T(sn)‖ sn = inf ‖T(sn)‖ sn 1

1

t→+∞

n→+∞

1

n≥1

for every s > 0. In this situation we can define the weighted space MωT = MωT [(0, +∞)] consisting of all regular measures μ on (0, +∞) such that the +∞ upper integral ∫0 ‖T(t)‖d|μ|(t) is finite. Since ωT (t1 + t2 ) ≤ ωT (t1 )ωT (t2 ) for t1 > 0, t2 > 0, we see that μ and ν are convolable for μ, ν ∈ MωT and that (MωT , ‖.‖ωT ) is a Banach algebra with respect to convolution which contains the convolution algebra Mc [(0, +∞)] of compactly supported regular measures on (0, +∞) as a dense subalgebra. Denote by L1ωT = L1ωT (R+ ) the convolution algebra of all Lebesgue-measurable (classes of) functions f on [0, +∞) such that fωT is integrable with respect to the Lebesgue measure on [0, +∞), identified to the space of all measures μ ∈ MωT which are absolutely continuous with respect to Lebesgue measure. Also denote by Cc∞ [(0, +∞)] the space of all infinitely differentiable functions f ∈ Cc [(0, +∞)]. We will say as usual that a sequence (f n )n≥1 of elements of +∞ L1 (R+ ) is a Dirac sequence if f n (t) ≥ 0 a.e. for t ≥ 0, ∫0 f n (t)dt = 1 and if there exists a sequence (a n )n≥1 of positive real numbers which converges to 0 such that f n (t) = 0 a.e. on [a n , +∞) for n ≥ 1. The following result is an easy analog of [2], Proposition 1.4. Proposition 2.4. Let (X, X∗ ) be a dual pairing, and let T = (T(t)t>0 ⊂ Bw (X) be a weakly continuous semigroup. The Pettis integral +∞

< ϕT (μ)x, l >= ∫ < T(t)x, l > dμ(t) (l ∈ X∗ )

(5)

0

defines for every x ∈ X and every μ ∈ MωT an element of X, ϕT (μ) ∈ B(X) for every μ ∈ MωT , and ϕT : μ 󳨃→ ϕT (μ) is a norm-decreasing unital algebra homomorphism from the convolution algebra MωT into B(X). Moreover if (f n )n≥1 is a Dirac sequence, then we have, for t > 0 (i) lim ‖ϕT (f n ∗ δ t )ϕT (f) − T(t)ϕT (f)‖ = 0 ∀f ∈ L1ωT , n→+∞

(ii) lim ⟨ϕT (f n ∗ δ t )x − T(t)x, l⟩ = 0 ∀x ∈ X, ∀l ∈ X∗ . n→+∞

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Proof. Since ‖T(t)x‖ ≤ ‖T(t)‖‖x‖, the fact that formula 5 defines an element of X for x ∈ X and μ ∈ MωT follows directly from Proposition 2.2. We have ‖ϕT (μ)x‖

󵄨󵄨 +∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 ∫ ⟨T(t)x, l⟩dμ(t)󵄨󵄨󵄨 󵄨󵄨 l∈X∗ ,‖l‖≤1 󵄨󵄨 󵄨0 󵄨

=

sup

+∞

≤

sup l∈X∗ ,‖l‖≤1

∫ |⟨T(t)x, l⟩| d|μ|(t) ≤ ‖x‖‖μ‖ωT , 0

and so ϕT (μ) ∈ B(X) for μ ∈ MωT and ‖ϕT (μ)‖ ≤ ‖μ‖ωT . As observed in [2], a routine application of Fubini’s theorem shows that ϕT (μ ∗ ν) = ϕT (μ)ϕT (ν) for μ, ν ∈ Mc ((0, +∞)). Since Mc ((0, +∞)) is dense in MωT , this shows that ϕT is an algebra homomorphism. The last assertions pertain to folklore, but we give the details for the sake of completeness. Recall that a sequence (e n )n≥1 of elements of a commutative Banach algebra A is said to be a sequential bounded approximate identity for A if supn≥1 ‖e n ‖ < +∞ and if limn→+∞ ‖ae n − a‖ = 0 for every a ∈ A. Let n ≥ 1, and let (f n )n≥1 be a Dirac sequence. We can assume without loss of generality that f n (t) = 0 a.e. on [1, +∞) for n ≥ 1. The sequence (f n )n≥1 is a sequential bounded approximate identity for L1 (R+ ). So if f ∈ L1ωT , and if supp(f) ⊂ [a, b], with 0 < a < b < +∞, we have lim sup ‖f n ∗ δ t ∗ f − δ t ∗ f‖L1 ≤ M lim sup ‖f n ∗ δ t ∗ f − δ t ∗ f‖L1 (R+ ) = 0, ωT

n→+∞

n→+∞

where M = supa+t≤s≤b+1+t ‖ωT (s)‖ < +∞. 1 Since ‖f n ∗ δ t ‖L1ω ≤ supt≤s≤t+1 ‖T(s)‖ ∫0 f n (s)ds = supt≤s≤t+1 ‖T(s)‖ for n ≥ 1, T a standard density argument shows that lim sup ‖f n ∗ δ t ∗ f − δ t ∗ f‖L1 = 0, n→+∞

ωT

and (i) holds. We have, for x ∈ X, l ∈ X∗ , t > 0

|< ϕT (f n ∗ δ t )x − T(t)x, l >|

󵄨󵄨 󵄨󵄨 a n an 󵄨󵄨 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨 ∫ < T(s + t)x, l > f n (s)ds− < T(t)x, l > ∫ f n (s)ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 0 an

≤ sup |< T(s + t)x − T(t)x, l >| ∫ f n (t)dt 0≤t≤a n

0

= sup |< T(s + t)x − T(t)x, l >| .

󵄨 󵄨 Hence limn→+∞ 󵄨󵄨󵄨< ϕT (f n ∗ δ t )x − T(t)x, l >󵄨󵄨󵄨 = 0, since the semigroup is weakly continuous with respect to the dual pairing (X, X∗ ). 0≤t≤a n

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We will use the following notations, for x ∈ X, μ ∈ MωT , +∞

+∞

0

0

ϕT (μ)x = ∫ T(t)xdμ(t), ϕT (μ) = ∫ T(t)dμ(t).

(6)

In particular we will write, for f ∈ L1ωT , +∞

ϕT (f) = ∫ f(t)T(t)dt.

(7)

0

3 The Arveson ideal Denote by X1 = [∪t>0 T(t)(X)]− the closure of the space ∪t>0 T(t)(X) in (X, ‖.‖). Since Mc ((0, +∞)) is dense in MT , we have ϕT (μ)x ∈ X1 for every μ ∈ MωT , x ∈ X. So we may consider ϕT as an element of B(X1 ) for every μ ∈ MωT , and ‖ϕT (μ)‖B(X) ≥ ‖ϕT (μ)‖B(X1 ) for every μ ∈ MT . The trivial example obtained by setting T(t) = P for t > 0, where P : X → Im(P) is a projection satisfying ‖P‖B(X) > 1, shows that equality does not hold in general. Definition 3.1. We will denote by AT the closed subalgebra of B(X1 ) generated by ϕT (MωT ), and by IT the closed subalgebra of B(X1 ) generated by ϕT (L1ωT ), so that IT is a closed ideal of AT , called the Arveson ideal of the semigroup T. Notice that AT is commutative. The fact that IT is an ideal of AT follows from the fact that L1ωT is an ideal of the convolution algebra MωT , so that ϕT (L1ωT ) is an ideal of ϕT (MωT ). Proposition 3.2. (i) The map t → T(t)u is norm-continuous on (0, +∞) for every u ∈ IT . (ii) The ideal IT equals the whole algebra AT if and only if the semigroup (T(t))t>0 is continuous with respect to the norm of B(X). (iii) If X∗ is closed in X 󸀠 , and if the σ(X∗ , X)-closed convex hull of every σ(X∗ , X)compact subset of X∗ is σ(X∗ , X)-compact, then all elements of AT are weakly continuous with respect to X∗ . Proof. The map t → δ t ∗ f is continuous on (0, +∞) for every f ∈ L1ωT . Hence the map t → T(t)ϕT (f) is continuous on (0, +∞) for every f ∈ L1ωT . Since supa≤t≤b ‖T(t)‖ < +∞ for 0 < a < b < +∞, this shows that (i) holds. Now assume that the semigroup is continuous with respect to the norm on B(X). A well-known argument, see for example Proposition 6.1 in [11], shows then

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that T(t) ∈ IT for every t > 0. Hence ϕT (μ) ∈ IT for every μ ∈ Mc ((0, +∞)). Since Mc ((0, +∞)) is dense in MωT , this shows that AT = IT . Conversely it follows immediately from (i) that if IT = AT then the semigroup ̃ (T(t))t>0 is continuous with respect to the norm of B(X). ∗ 󸀠 Now assume that X is closed in X , and that the σ(X∗ , X)-closed convex hull of every σ(X∗ , X)-compact subset of X∗ is σ(X∗ , X)-compact. Then (X∗ , X) is a dual pairing, and we see as in the proof of the second assertion of proposition 1.4 of [2] that l ∘ ϕT (μ) ∈ X∗ for μ ∈ MωT , l ∈ X∗ . Hence l ∘ u ∈ X∗ for u ∈ AT , l ∈ X∗ . Proposition 3.3. Let u ∈ AT . If uv = 0 for every v ∈ IT , then u = 0.

Proof. Let (f n )n≥1 be a Dirac sequence, and let ϵ n be a sequence of positive real numbers such that limn→+∞ ϵ n = 0. We have, for x ∈ X, l ∈ X∗ , t > 0, < uT(t)x, l >=< T(t)ux, l >= lim < ϕT (f n ∗ δ ϵ n ∗ δ t )ux, l >= n→+∞

= lim < uϕT (f n ∗ δ ϵ n )T(t)x, l >= 0. n→+∞

Hence uT(t)x = 0 for every x ∈ X, and so u = 0 since ∪t>0 T(t)X is dense in X1 .

Proposition 3.4. If lim supt→0+ ‖T(t)‖ < +∞, then the sequence (ϕT (f n ))n≥1 is a bounded sequential approximate identity for IT for every Dirac sequence (f n )n≥1 . Proof. Assume that lim supt→0+ ‖T(t)‖ < +∞, and set M = sup00 as operators on the Banach space X1 := [∪t>0 T(t)X]− . We set X̃ := {x ∈ X1 | lim < T(t)x − x, l >= 0 ∀l ∈ X∗ }. t→0+

We will use the convention T(0) = I X1 , the identity map on X1 .

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Clearly, ∪t>0 T(t)X ⊂ X,̃ and [∩t>0 Ker(T(t)] ∩ X̃ = {0}. Notice that if x ∈ X,̃ it follows from the Banach-Steinhaus theorem that lim sup ‖T(t)x‖ = lim sup ‖T(t)x‖(X∗ )󸀠 < +∞. t→0+

t→0+

Also if lim supt>0 ‖T(t)‖ < +∞, then X̃ is closed, so that X̃ = X1 . The following definition is a variant of a definition introduced by Feller in [17] for semigroups satisfying some weak measurability conditions. Definition 4.1. Denote by DT,op the space of all x ∈ X1 such that there exists y ∈ X̃ satisfying lim
= 0 ∀l ∈ X∗ . t

(8)

In this situation we set AT,op x = y.

Notice that if x ∈ DT,op , then AT,op x ∈ X1 . Also l ∘ T(t) ∈ X∗ for every t > 0 and every l ∈ X∗ , and so when condition (8) is satisfied by x and y we have lim
= 0 (l ∈ X∗ , s > 0). t

If x ∈ DT,op , and if y = AT,op x, then we have, for l ∈ X∗ , t > 0, t

< T(t)x − x, l >= ∫ < T(s)y, l > ds, 0

and so, according to the notations of formula (4), we have t

T(t)x − x = ∫ T(s)yds.

(9)

0

t Conversely, if y ∈ X,̃ and if T(t)x − x = ∫0 T(s)yds for t > 0, then x and y obviuously satisfy (8) and so x ∈ DT,op , and y = AT,op x. 1 Now set ρT = limt+∞ ‖T(t)‖ t , and for Re(λ) > log(ρT ), y ∈ X,̃ define R op (T, λ)y by the formula +∞

< R op (T, λ)y, l >:= ∫ e−λs < T(s)y, l > ds ∀l ∈ X∗ . 0

In other terms, according to the notations of formula (6), we have +∞

R op (T, λ)y = ∫ e−λs T(s)yds. 0

(10)

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Proposition 4.2. If y ∈ X,̃ then R op (T, λ)y ∈ DT,op , R op (T, λ) : X̃ → DT,op is one-to-one and onto, and we have (λI X1 − AT,op ) ∘ R op (T, λ) = I X̃ , R op (T, λ) ∘ (λI X1 − AT,op ) = IDT,op .

Proof. The proof is standard, and we give the details for the sake of completeness. +∞ Let y ∈ X.̃ Since ∫0 e−Re(λ)s ‖T(s)y‖ds < +∞, R op (T, λ)y is well-defined. Set then x = R op (T, λ)y. Since l ∗ T(t) ∈ X∗ for every l ∈ X∗ and every t > 0, we have +∞

+∞

0

t

T(t)x = ∫ e−λs T(t + s)yds = e λt ∫ e−λs T(s)yds, ∫ e e λt − 1 T(t)x − x = x − e λt 0 t t t

−λs T(s)yds

t

,

where the integral is computed with respect to the weak operator topology on X associated to the dual pairing (X, X∗ ). We obtain, for l ∈ X∗ , lim
=< λx − y, l >, t

and so x ∈ DT,op , and AT,op x = λx − y, y = λx − AT,op x.

This shows that R op (T, λ)(X)̃ ⊂ DT,op , and that (λI X1 − AT,op )∘ R op (T, λ) = I X̃ , and so RT,op (λ) : X̃ → DT,op is one-to-one. Conversely let x ∈ DT,op , set u = AT,op x, and set S(t) = e−λt T(t) for t > 0. Then x ∈ DS,op , and if we set y = AS,op x we have y = u − λx. We have, for t > 0, t

t

0

0

S(t)x − x = ∫ S(s)yds = ∫ e−λs T(s)yds.

Since limt→+∞ ‖S(t)‖ = 0, we obtain +∞

x = − ∫ e−λs T(s)yds = −RT,op (λ)y = λRT,op (λ)x − (RT,op (λ) ∘ AT,op )x. 0

Hence RT,op (λ) : X̃ → DT,op is onto, and we have RT,op (λ) ∘ (λI X1 − AT,op ) = IDT,op .

Proposition 4.3. The domain DT,op of AT,op is weakly dense in X1 .

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Proof. Let f ∈ Cc∞ ([0, +∞)), set m := supt∈supp(f) ‖T(t)‖, and denote by ‖.‖1 the usual norm on L1 (R+ ). Then 󵄩󵄩 δ ∗ f − f 󵄩󵄩 󵄩󵄩 󵄩󵄩 δ ∗ f − f 󵄩 󵄩 󵄩 t 󵄩 t ≤ m lim 󵄩󵄩󵄩 − f 󸀠 󵄩󵄩󵄩 − f 󸀠 󵄩󵄩󵄩 = 0, lim 󵄩󵄩󵄩 󵄩󵄩L1ω 󵄩󵄩1 󵄩󵄩 t t t→0+ 󵄩 󵄩 T

t→0+

and so

󵄩󵄩 T(t) ∗ ϕ (f) − ϕ (f) 󵄩󵄩 T T 󵄩 󵄩 lim 󵄩󵄩󵄩 − ϕT (f 󸀠 )󵄩󵄩󵄩 = 0. + 󵄩 󵄩󵄩 t t→0 󵄩

Since ϕT (f 󸀠 ) ⊂ T(t)AT for some t > 0, we have ϕT (f)(X) ⊂ X,̃ and so ϕT (f)(X) ⊂ DT,op . Now let (f n )n≥1 ⊂ Cc∞ ([0, +∞)) be a Dirac sequence. It follows from Proposition 2.4 that we have for x ∈ X, l ∈ X∗ , t > 0, lim < ϕT (f n )T(t)x − T(t)x, l >= 0.

n→+∞

Hence DT,op is weakly dense in ∪t>0 T(t)(X), and so DT,op is weakly dense in X1 .

Recall that a partially defined operator S : D → E on a Banach space E is said to be closed if the graph {x, Sx : x ∈ D} of S is closed. Note that if lim supt→0+ ‖T(t)‖ < +∞, then X̃ = X1 , and so R(T, λ) : X̃ → DT,op ⊂ X̃ is a bounded operator. Hence its inverse λI X̃ − AT,op : DT,op → X̃ is closed, and AT,op is also closed. Now assume that (X∗ , X) is also a dual pair, which means that X∗ is closed in X 󸀠 and that the σ(X∗ , X)-closed convex hull of every σ(X∗ , X)compact subset of X∗ is σ(X∗ , X)-compact. Set u λ (t) = e−λt for λ ∈ C , t ≥ 0. Then RT,op (λ) = ϕT (u λ ) for Re(λ) > ρT , and so it follows from Proposition 3.2(iii) that RT,op (λ) is weakly continuous with respect to X∗ , which shows as above that AT,op is weakly closed.

5 Quasimultipliers on the Arveson ideal and Arveson spectrum If U be a commutative Banach algebra, set Ω(U) := {u ∈ U | [uU]− = U}, and set U⊥ = {u ∈ U | uU = {0}}. We recall the definition of quasimultipliers, a notion introduced by the author in [12]. Definition 5.1. [12] Assume that Ω(U) ≠ 0 and that U⊥ = {0}. A quasimultiplier on U is a pair (S u/v , DS u/v ), where u ∈ U, v ∈ Ω(U), where DS u/v is the ideal of U consisting of those x ∈ U such that ux ∈ vU, and where S u/v x is the unique y ∈ U such that ux = vy for x ∈ DS u/v . The set of quasimultipliers on U will be denoted

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by QM(U), and a set B ⊂ QM(U) will be said to be pseudobounded if there exists x ∈ [∩S∈B DS ] ∩ Ω(U) such that supS∈B ‖Sx‖ < +∞. The quasimultipliers on U form an algebra, which is isomorphic to the algebra of fractions U/Ω(U). Notice that since Ω(U) is stable under products, the product of two pseudobounded sets is pseudobounded, which gives QM(U) a structure of "algèbre à bornés". The algebra QM(U) is in some sense too large, since u is invertible in QM(U) for every u ∈ Ω(U), so it is natural to consider the following class.

Definition 5.2. [12] A quasimultiplier R ∈ QM(U) is said to be regular if there exists λ > 0 such that the family (λ n R n )n≥1 is pseudobounded. The set of regular quasimultipliers on U will be denoted by QM r (U), and a set B ⊂ QMr (U) will be said to be multiplicatively pseudobounded if it is contained in some set of the form λV, where λ > 0 and where V is a pseudobounded subset of QMr (U) which is stable under products.

Recall that a multiplier on U is a bounded linear map S : U → U such that (Su)v = S(uv) for u, v ∈ U. The set M(U) of multipliers on M(U) is a closed subalgebra of B(U), and U can be identified to an ideal of M(U) in an obvious way. We also have the obvious identification M(U) := {S = S u/v ∈ QM(U) | DS u/v = U} ⊂ QMr (U). Equipped with the family of all its multiplicatively pseudobounded subsets, the set QMr (U) forms an algebra which is a pseudo-Banach algebra in the sense of Allan, Dales and McClure [1]. In particular every maximal ideal of QMr (U) is the kernel of a character of QMr (U), and QMr (U) is an inductive limit of commutative Banach algebras. In fact regular quasimultipliers on U can be turned into multipliers in the usual sense by modifying the algebra U, as will be seen later. Precise estimates on inverse Laplace transforms, obtained by the author in section 6 of [11] in collaboration with Paul Koosis, combined with Nyman’s characterization of dense ideals of L1 (R+ ), show that if ω is a positive, continuous, nonincreasing submultiplicative weight on (0, +∞) the convolution algebra L1ω := L1ω (R+ ) contains an infinitely differentiable convolution semigroup (a(t))t>0 such that a(t) ∗ L1ω is dense in L1ω for every t > 0, [11], theorem 6.8. This result uses the fact that if f is a continuous function on (0, +∞) such that lim supt→∞ f(t) < +∞, then there exists a continuously differentiable function h on (0, +∞) such that h(t) ≥ f(t) for t > 0, and the rather sophisticated contruction, applied to the function f : t → e−λt ω(t) for some suitable λ ∈ R , consists in finding a convolution semigroup (a(t))t>0 of functions in L1 (R+ ) such that lim supt→0+ h(t)‖a(t)‖ < +∞ and such that the convolution ideal generated by a(t) is dense in L1 (R+ ) for every

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t > 0 (which implies that the Laplace tranform of a(t) is an outer function of the right-hand half-plane). The fact that a(t) ∗ L1ω is dense in L1ω follows from Cohen’s factorization theorem for modules over commutative Banach algebras with bounded approximate identities, which implies that every f ∈ L1ω can we written under the form f = g ∗ h, where g ∈ L1 (R+ ) and h ∈ L1ω . Now assume that f is positive and lower semicontinuous on (0, +∞), so that f is bounded on [a, b] for 0 < a < b < +∞, and that lim supt→+∞ f(t) < +∞. t Set f1 (t) = sups≥t f(s), f2 (t) = 2t ∫t/2 f1 (s)ds. Then f1 is nonincreasing, f1 (s/2) ≥ f2 (s) ≥ f1 (s) ≥ f(s), and applying the previous result to f2 we see that there exists a continuously differentiable function h on (0, +∞) such that h(t) ≥ f(t) for t > 0. This shows that the construction used in the proof of theorem 6.8 of [11] extends to lower-semicontinuous weights on (0, +∞). Set u λ (t) = e λt for t > 0. Since the map f → fu−λ is a norm-preserving homomorphism from the convolution algebra L1ω (R+ ) onto the convolution algebra L1u λ ω (R+ ) for λ ∈ R , we obtain the following result. Theorem 5.3. Let ω be a positive, lowersemicontinuous, submultiplicative weight on (0, +∞), and assume that ω̃ : t → e λt ω(t) is nonincreasing for some λ ∈ R . Then the convolution algebra L1ω := L1ω (R+ ) contains an infinitely differentiable semigroup (a(t))t>0 such that a(t) ∗ L1ω is dense in L1ω for every t > 0. Set again ρT := limt→+∞ ‖T(t)‖ t . We obtain the following result, which shows in particular that the ideal IT possesses dense principal ideals, and that the family (e−λt T(t))t>0 is pseudobounded in QM(IT ) for λ > ρT . 1

Proposition 5.4. Let λ > log (ρT ), and set ω λ (t) = e λt sups≥t e−λs ‖T(s)‖. Then ‖ϕT (g)T(t)‖ ≤ e λt ‖g‖L1ω for every g ∈ L1ω λ and every t > 0, and ϕT (g) ⊂ Ω(IT ) for every g ∈ Ω (L1ω λ ) .

λ

Proof. Notice that ‖T(t)‖ ≤ ω λ (t) for t > 0, so that L1ω λ ⊂ L1ωT . Also ω λ (s) ≥ e λs supt≥0 e−λ(s+t) ‖T(s + t)‖, and so ‖T(s + t)‖ ≤ e λt ω λ (s) for s ≥ 0, t > 0. We have, for t > 0, g ∈ L1ω λ , 󵄩󵄩 +∞ 󵄩󵄩 +∞ 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ‖ϕT (g)T(t)‖ = ‖ϕT (g ∗ δ t )‖ = 󵄩󵄩󵄩 ∫ g(s − t)T(s)ds󵄩󵄩󵄩 ≤ ∫ |g(s − t)|‖T(s)‖ds 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 t t +∞

≤ ∫ |g(s)|‖T(s + t)‖ds ≤ e λt ‖g‖L1 . ω λ

0

1 1 1 1 Since L1ω λ contains C∞ c [(0, +∞)], L ω λ is dense in L ωT , and so Ω(L ω λ ) ⊂ Ω(L ωT ), 1 which shows that ϕT (Ω(L ω λ )) ⊂ Ω(IT ).

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Denote by ̂ IT the space of characters of IT , equipped with the usual Gelfand topology. Notice that if χ ∈ ̂ IT then there exists a unique character χ̃ on QM(IT ) such ̃ ̃ u/v ) = χ(u) that χ|IT = χ, which is defined by the formula χ(S χ(v) for u ∈ IT , v ∈ Ω(IT ). This leads to the following natural notion, to be studied in section 9. Definition 5.5. Assume that IT is not radical, and let S ∈ QM(IT ). The Arveson spectrum σ ar (S) is defined by the formula ̃ σ ar (S) = {λ = χ(S) : χ∈̂ IT }.

6 Quasimultipliers on the Arveson ideal as weakly densely defined operators We now associate to every quasimultiplier on IT a weakly densely defined operator on X1 . Set again X̃ := {x ∈ X1 | limt→0+ < T(t)x − x, l >= 0 ∀l ∈ X∗ }. Lemma 6.1. Let v ∈ Ω(IT ). Then Ker(v) ⊂ ∩t>0 Ker(T(t)), and v : X̃ → X1 is one-to-one.

Proof. Let x ∈ Ker(v), and let (f n )n≥1 ⊂ Cc ((0, ∞)) be a Dirac sequence. Since vIT is dense in IT , there exists a sequence (w n )n≥1 of elements of IT such that limn→+∞ ‖ϕT (f n ) − vw n ‖ = 0. It follows then from proposition 2.4 (ii) that we have, for x ∈ X, l ∈ X∗ , t > 0, lim n→+∞ < vT(t)w n x − T(t)x, l >= 0.

Since vT(t)w n x = T(t)w n vx = 0, we have x ∈ ∩t>0 Ker(T(t)). This shows that v is one-to-one on X,̃ since (∩t>0 Ker(T(t)) ∩ X̃ = {0}.

̃ and Proposition 6.2. Let S = S u/v ∈ QM(IT ). Set DS̃ := {x ∈ X | ux ∈ v(X)}, ̃ ̃ denote by S(x) the unique y ∈ X satisfying ux = vy. Then DS̃ is weakly dense in X1 . If lim supt→0+ ‖T(t)‖ < +∞, then S̃ is a closed partially defined operator on X1 , and S̃ is weakly closed if, further, the σ(X∗ , X)-closed convex hull of every σ(X∗ , X)compact subset of X∗ is σ(X∗ , X)-compact.

Proof. It follows from the lemma that S̃ is well-defined on DS̃ . Since vIT is dense in IT , we see as above there exists a sequence (w n )n≥1 of elements of IT such that we have, for x ∈ X, l ∈ X∗ , t > 0, lim < vT(t)w n x − T(t)x, l >= 0.

n→+∞

̃ we have vT(t)w n x ∈ D ̃ for t > 0, n ≥ Since uvT(t)w n = vuT(t)w n ∈ v(X), S 1, x ∈ X, and we see that the closure of DS̃ in X1 with respect to the weak topology

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σ(X, X∗ ) contains T(t)(X) for every t > 0, which shows that DS̃ is weakly dense in X1 . If lim supt→0+ ‖T(t)‖ < +∞, then X̃ = X1 , and DS̃ = {x ∈ X1 | ux ∈ v(X1 )}. Let (x, y) ∈ X1 × X1 , and assume that there exists a sequence (x n )n≥1 of elements of ̃ n ‖ = 0. DS̃ such that limn→+∞ ‖x − x n ‖ =limn→+∞ ‖y − Sx ̃ n = vy. Hence x ∈ D ̃ and y = Sx, ̃ Then ux =limn→+∞ ux n =limn→+∞ v Sx S ̃ which shows that S is a closed operator. Now if the σ(X∗ , X)-closed convex hull of

every σ(X∗ , X)-compact subset of X∗ is σ(X∗ , X)-compact, then u and v are weakly continuous, and a similar argument using generalized sequences shows that S̃ is weakly closed. Assume that ∩t>0 Ker(T(t)) = {0}. Then v is one-to-one on X1 for every v ∈ Ω(IT ). If S = S u/v ∈ QM(IT ), then we can set DS̃ := {x ∈ X1 |ux ∈ v(X1 )},

(11)

̃ = ux (x ∈ D ). v(Sx) S̃

(12)

, and define S̃ : D ̃ by the formula S

Then S̃ is a closed extension of S.̃ Notice that if lim supt→0+ ‖T(t)‖ < +∞, then S̃ is also weakly closed. To deal with the general case observe that if x ∈ X1 , and if T(s)x ∈ ∩t>0 T(t)(X)

for every s > 0, then T(t)(x) = T(t/2)(T(t/2)x) = 0 for every t > 0. So if we set X0 := X1 / ∩t>0 Ker(T(t)), and if we set ̃ T(t)(x + ∩t>0 Ker(T(t)) = T(t)x + ∩t>0 Ker(T(t))

̃ for x ∈ X1 , t > 0, then ∩t>0 Ker(T(t)) = {0}. Now if u ∈ AT , then u(∩t>0 Ker(T(t)) ⊂ ∩t>0 Ker(T(t),

and we can define ũ : X0 → X0 by the formula ̃ + ∩t>0 Ker(T(t)) = ux + ∩t>0 Ker(T(t)) (x ∈ X1 ), u(x

and the same argument as in the proof of Lemma 6.1 shows that if v ∈ Ω(IT ), and if vx ∈ ∩t>0 Ker(T(t)) for some x ∈ X1 , then x ∈ ∩t>0 Ker(T(t)), so that ṽ is one-to-one on X0 . This suggests the following definition. ̃ ∈ v(X ̃ 0 )}. The Definition 6.3. Let S = S u/v ∈ QM(IT ). Set DS̃ := {y ∈ X0 / | uy operator S̃ : DS̃ → X0

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is defined by the formula

̃ = uy, ̃ (y ∈ D ̃ ). ṽ (Sy) S

We see that S̃ is a closed operator on X0 , which may be considered as an extension of S̃ since (∩t>0 Ker(T(t))) ∩ X̃ = {0}.

7 The normalized Arveson ideal It follows from Theorem 5.3 and Proposition 5.4 that the family (e−λt T(t))t>0 is pseudobounded in QM(IT ) for λ > log(ρT ). We now apply to the Arveson ideal IT and to the family (T(t))t>0 a slight modification of construction performed by the author in [12] to embed a pseudobounded subset U stable under products of the quasimultiplier algebra QM(A) of a Banach algebra A such that uA ≠ {0} for u ∈ A \ {0} and Ω(A) ≠ 0 into a bounded set of the multiplier algebra of some Banach algebra B which is "similar" to A. In order to do this, we start by setting L := {u ∈ A | ‖u‖L := supb∈U ‖bu‖ < +∞}, choose b ∈ L ∩ Ω(A), and denote by U the closure of bL in (L, ‖.‖L ); then b2 ∈ Ω(U). The algebra B introduced in [12] is the closure of A in M(L), the map S u/v → S b2 u/b2 v defines a pseudobounded isomorphism from QM(B) onto QM(A) and QM(U), and the given pseudobounded set U is contained in the unit ball of M(B). This construction was inspired by Feller’s discussion [18] of the generator of strongly continuous semigroups which are not bounded near the origin, and we slightly modify it below by replacing the multiplier algebra M(L) by the multiplier algebra M(U) in order to obtain a norm-preserving isomorphism between the multiplier algebras M(U) and M(B) in the case where U = {e−λt T(t)} for some sufficiently large real number λ. We will use the convention T(0)u = u for u ∈ IT .

Theorem 7.1. Set LT := {u ∈ IT | lim supt→0+ ‖T(t)u‖ < +∞} ⊃ ∪t>0 T(t)IT , choose λ > log(ρT ), set ‖u‖λ := sups≥0 e−λs ‖T(s)u‖ for u ∈ LT , denote by UT the closure of ∪t>0 T(t)IT in (LT , ‖.‖λ ), and set ‖R‖λ,op := sup{‖Ru‖λ | u ∈ UT , ‖u‖λ ≤ 1} = ‖R‖M(UT ) for R ∈ M(UT ). Then (LT , ‖.‖λ ) is a Banach algebra, the norm topology on LT does not depend on the choice of λ, UT = {u ∈ LT | limt→0+ ‖T(t)u − u‖λ = 0}, UT is a dense ideal of IT , and the following properties hold: (i) ‖T(t)u‖λ ≤ (sups≥0 e−λs ‖T(t + s)‖) ‖u‖ for u ∈ IT , t > 0. (ii) ‖T(t)‖λ,op ≤ e λt for every t > 0. (iii) R(LT ) ⊂ LT and R(UT ) ⊂ UT for every R ∈ M(IT ), and ‖R‖λ,op ≤ ‖R‖M(IT ) . In particular ‖u‖λ,op ≤ ‖u‖ for every u ∈ AT . (iv) ab ∈ UT for a ∈ LT , b ∈ LT .

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(v) Ω(IT )∩LT ≠ 0, aLT is dense in UT for every a ∈ Ω(IT )∩LT , and ab ∈ Ω(UT ) for a ∈ Ω(IT ) ∩ LT , b ∈ Ω(IT ) ∩ LT . Now identify IT to a subset of M(UT ), and denote by JT the closure of IT in (M(UT ), ‖.‖λ,op ). Then UT is a dense ideal of (JT , ‖.‖λ,op ) and the following properties hold: (vi) (ϕT (f n ∗ δ ϵ n ))n≥1 is a bounded sequential approximate identity for JT for every Dirac sequence (f n )n≥1 and every sequence (ϵ n )n≥1 such that limn→+∞ ϵ n = 0, and lim supn→+∞ ‖ϕT (f n ∗ δ ϵ n )‖λ,op = 1. (vii) Ru ∈ UT for every R ∈ M(JT ) and every u ∈ UT , and the map R → R|UT is a norm-preserving isomorphism from M(JT ) onto M(UT ). (viii) If a ∈ Ω(UT ) then the map S u/v → S au/av defines a pseudbounded isomorphism from QM(JT ) onto QM(IT ) and from QM(JT ) onto QM(UT ). The Banach algebra JT , which is a closed ideal of its multiplier algebra M(JT ), will be called the normalized Arveson ideal of the semigroup T. Proof. Let λ > log(ρT ), and fix a > 0. Set k(a) = sups≥a e−λs ‖T(s)‖. Property (i) follows from the definition of the norm ‖.‖λ , and we have ‖u‖λ ≤ max( sup ‖e−λs T(s)u‖, k(a)‖u‖) ≤ max(e|λ|a sup ‖T(s)u‖, k(a)‖u‖) 0≤s≤a

0≤s≤a

≤ max(e|λ|a , k(a)) sup ‖T(s)u‖, 0≤s≤a

sup ‖T(s)u‖ ≤ e|λ|a sup e−λs ‖T(s)u‖ ≤ e|λ|a ‖u‖λ .

0≤s≤a

0≤s≤a

Hence all the norms ‖.‖λ are equivalent to the norm u → sup0≤s≤a ‖T(s)u‖. The construction of LT is the first part of the construction of Theorem 7.11 of [12] applied to the pseudobounded family stable under products {e−λt T(t)}t>0 . It follows then from Theorem 7.11 of [12] that (LT , ‖.‖λ ) is a Banach algebra, and (ii) follows from the fact that ‖T(t)u‖λ ≤ e λt ‖u‖λ for u ∈ LT , t > 0. It also follows from theorem 7.11 of [12] that LT is an ideal of M(IT ) ⊃ AT and that we have the obvious inequality ‖R‖op,λ ≤ ‖R‖M(IT ) for every R ∈ M(IT ), which implies that LT is an ideal of AT and that ‖a‖op,λ ≤ ‖a‖ for every a ∈ AT . This implies that R(UT ) ⊂ UT for every R ∈ M(IT ), which completes the proof of (iii). Set q(u) = sup0≤s≤1 ‖T(s)u‖ for u ∈ LT , so that the norm q is equivalent to the norm ‖.‖λ on LT . Let t > 0 and let u ∈ IT . Set k := supt/2≤s≤(t/2)+1 ‖T(s)‖. Then q(T(t + r)u − T(t)u) ≤ k‖T(t/2 + r)u − T(t/2)u‖

and so lim supr→0+ q(T(r)T(t)u − T(t)u) = 0.

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It follows then from (ii) that UT ⊂ {u ∈ LT | limt→0+ ‖T(t)u − u‖λ = 0}, and in UT = {u ∈ LT | lim+ ‖T(t)u − u‖λ = 0}, t→0

since the other inclusion is trivial. Let a ∈ LT , b ∈ LT . There exists a sequence (u n )n≥1 of elements of ∪t>0 T(t)IT such that limn→+∞ ‖b − u n ‖ = 0, and we have lim sup ‖ab − au n ‖λ ≤ ‖a‖λ lim sup ‖b − u n ‖ = 0. n→+∞

n→+∞

Hence ab ∈ UT , which proves (iv). Now let a ∈ Ω(IT ) ∩ LT . Then aLT ⊂ UT . Let u ∈ IT , and let (u n )n≥1 be a sequence of elements of IT such that limn→+∞ ‖u − au n ‖ = 0. It follows from (i) that limn→+∞ ‖T(t)u − T(t)au n ‖λ = 0, and so the closure of aLT in (UT , ‖.‖λ ) contains ∪t>0 T(t)IT , hence coincides with UT . An easy verification given in the proof of Theorem 7.11 of [12] shows then that ab ∈ Ω(UT ) for a ∈ Ω(IT ) ∩ LT , b ∈ Ω(IT ) ∩ LT . Notice that the ideals LT and UT play with respect to IT and the family (e−λt T(t))t>0 the role played by the ideals L and U with respect to the Banach algebra A and the pseudobounded family U stable under products in the construction of Theorem 7.11 of [12]. Now identify IT to a subset of M(UT ), and denote by JT the closure of IT in (M(UT ), ‖.‖λ,op ). By construction, UT is an ideal of JT which is dense in JT since it contains ∪t>0 T(t)IT . Then the Banach algebras UT , IT and JT are similar in the sense of [12], Definition 7.4 and the natural injection IT → JT in a s-homomorphism in the sense of [12], Definition 7.9, and it follows from Theorem 7.11 of [12] that (viii) holds. Let (f n )n≥1 be a Dirac sequence and let (ϵ n )n≥1 be a sequence of positive real numbers such that limn→+∞ ϵ n = 0. There exists a sequence (r n )n≥1 of positive real numbers such that limn→+∞ r n = 0 and such that f n (t) = 0 a.e. for t > r n . Set e n = ϕT (f n ∗ δ ϵ n ). We have rn

‖e n ‖λ,op ≤ ∫ f n (t)‖T(t + ϵ n )‖λ,op dt, 0

and it follows from (ii) that lim supn→+∞ ‖e n ‖λ,op ≤ 1. Since limn→+∞ ‖e n u−u‖ = 0, we have a fortiori limn→+∞ ‖e n u−u‖λ = 0 for u ∈ ∪t>0 T(t)IT , and a standard density argument shows that limn→+∞ ‖e n u − u‖λ = 0 for u ∈ JT . Hence (e n )n≥1 is a sequential approximate identity for JT and satisfies lim n→+∞ ‖e n ‖op,λ = 1. This implies that ‖u‖M(JT ) = ‖u‖λ,op for u ∈ JT . Let u ∈ JT , let S ∈ M(UT ). Since UT is dense in (IT , ‖.‖), UT is dense in IT with respect to ‖.‖λ,op , and there exists a sequence (u n )n≥1 of elements of UT such that limn→+∞ ‖u−u n ‖λ,op = 0. Then limn→+∞ ‖Su−Su n ‖λ,op = 0, which shows that Su ∈ JT . Hence S : UT → UT has an extension S̃ to JT , and ‖S‖̃ M(JT ) = ‖S‖λ,op .

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Writing T(t)u = T(t/2)T(t/2)u for t > 0, u ∈ IT , we deduce from Proposition 2.4 and (ii) that limn→+∞ ‖T(t)u − T(t)ue n ‖λ = 0. A standard density argument shows then that lim ‖ue n − u‖λ = 0

n→+∞

for every u ∈ UT . It follows then from Cohen’s factorization theorem [8] for Banach modules over commutative Banach algebras with bounded approximate identities that for every u ∈ UT there exists v ∈ JT and w ∈ UT such that u = vw, ‖v‖λ,op = 1 and ‖w‖λ ≤ ‖u‖λ + ϵ. Hence Ru = (Rv)w ∈ UT for R ∈ M(JT ). Also ‖Ru‖λ ≤ ‖Rv‖λ,op ‖‖w‖λ ≤ ‖R‖M(JT ) (‖u‖λ + ϵ)

for every ϵ > 0, ‖Ru‖λ ≤ ‖R‖M(UT ) ‖u‖λ , which shows that RłUT : UT → UT is bounded. So the map R → R|UT is a norm-preserving isomorphism from M(JT ) onto M(UT ).

8 The generator as a quasimultiplier on the Arveson ideal When S = S u/v , with u ∈ A, v ∈ Ω(A) is a quasimultiplier on a commutative Banach algebra A such that Ω(A) ≠ 0 and uA ≠ {0} for u ∈ A \ {0}, we will write S = u/v if there is no risk of confusion. Definition 8.1. : The infinitesimal generator AT,IT of an Arveson weakly continuous semigroup T is the quasimultiplier on IT defined by the formula AT,IT = −ϕT (f0󸀠 )/ϕT (f0 ),

where f0 ∈ C 1 ([0, +∞)) ∩ Ω (L1ωT ) satisfies f0 (0) = 0, f0󸀠 ∈ L1ωT .

Assume that f1 also satisfies the conditions of the definition, and set f2 = f0 ∗ f1 . Since Ω (L1ωT ) is stable under convolution, f2 ∈ Ω (L1ωT ) , and f2󸀠 = f0󸀠 ∗ f1 = f0 ∗ f1󸀠 belongs to L1ωT , and f2󸀠 is continuous. Also f2 (0) = 0, and we have ϕT (f2󸀠 )/ϕT (f2 ) = ϕT (f0󸀠 )ϕT (f1 )/ϕT (f0 )ϕT (f1 ) = ϕT (f0󸀠 )/ϕT (f0 ),

and similarly ϕT (f2󸀠 )/ϕT (f2 ) = ϕT (f1󸀠 )/ϕT (f1 ), which shows that the definition of AT,IT does not depend on the choice of f0 . Let λ > log(ρT ), and set again u λ (t) = e λt and ω λ (t) = e λt sups≥t e−λs ‖T(s)‖ for t > 0. Then L1u λ ∗ L1ω λ ⊂ L1ω λ , and every Dirac sequence is a sequential bounded approximate identity for L1u λ . It follows from Cohen’s factorization theorem [8] that every f ∈ L1ω λ can be written under the form f = g ∗ h, where g ∈ L1u λ and h ∈ L1ω λ , which implies that Ω(L1u λ ) ∩ L1ω λ ⊂ Ω(L1ω λ ). It is well-known that the subalgebra of the convolution algebra L1 (R+ ) generated by u α is dense in L1 (R+ ), for α < 0, see for example [11], so that u α ∈ Ω(L1 (R+ )

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(this follows also from Nyman’s theorem [22]). So the function u α ∗ u α : t → te αt generates a dense ideal of L1 (R+ ), and if log(ρT ) < μ < λ then the function v λ : t → te−λt generates a dense ideal of L1u μ . Notice that v λ ∈ C ∞ ([0, +∞)), that v󸀠λ ∈ L1u μ and that v λ (0) = 0. Using proposition 5.4, we obtain the following result. Proposition 8.2. : Let μ > log(ρT ), let g0 ∈ C 1 ([0, +∞)) ∩ Ω (L1u μ ) such that

g0 (0) = 0, g0󸀠 ∈ L1u μ , and let h0 ∈ L1ω μ ∩ Ω (L1u μ ) . Then ϕT (g0 ∗ h0 ) ∈ Ω(IT ), and AT,IT = −ϕT (g0󸀠 ∗ h0 )/ϕT (g0 ∗ h0 ).

For example we can take g0 = v λ for some λ > log(ρT ), so that g0 (t) = te−λt and g0󸀠 (t) = (1 − λt)e−λt for t ≥ 0. We can also consider the infinitesimal generator of the semigroup as a quasimultiplier on the normalized Arveson ideal JT . Set ω̃ T (t) := ‖T(t)‖M(JT ) for all t > 0, and for μ ∈ Mω̃ T , define ϕ̃ T (μ) ∈ M(JT ) by the formula +∞

ϕ̃ T (μ)u = ∫ T(t)udμ(t) (u ∈ JT ).

(13)

0

Clearly, ϕ̃ T (μ)u = ϕT (μ)u for u ∈ IT and μ ∈ MωT , and so we can consider ϕ̃ T : Mω̃ T → M(JT ) as an extension to Mω̃ T of ϕT : MωT → M(IT ). Definition 8.3. The normalized infinitesimal generator AT,JT of a Arveson weakly continuous semigroup T is the quasimultiplier on JT defined by the formula AT,JT = −ϕ̃ T (g0󸀠 )/ϕ̃ T (g0 ),

where g0 ∈ C 1 ([0, +∞)) ∩ Ω (L1ω̃ T ) satisfies g0 (0) = 0, g0󸀠 ∈ L1ω̃ T .

Let w ∈ Ω(UT ). The map S u/v → S wu/wv defines a pesudobounded isomorphism from QM(JT ) onto QM(IT ). Since ϕ̃ T (g0 )ϕT (h0 ) = ϕT (g0 ∗h0 ) and ϕ̃ T (g0󸀠 )ϕT (h0 ) = ϕT (g0󸀠 ∗ h0 ) if g0 and h0 satisfy the conditions of proposition 8.2, the infinitesimal generator AT,IT is the quasimultiplier on IT associated to AT,JT via this isomorphism. Lemma 8.4. Let ω be a lower semicontinuous submultiplicative weight on (0, +∞), and let f ∈ C 1 ([0, +∞)) ∩ L1ω . If f(0) = 0, and if f 󸀠 ∈ L1ω , then the Bochner integral +∞ ∫t (f 󸀠 ∗ δ s )ds is well-defined in L1ω for t ≥ 0, and we have t 󵄩󵄩 f ∗ δ − f 󵄩󵄩 t 󵄩 󵄩 + f 󸀠 󵄩󵄩󵄩 = 0. f ∗ δ t − f = − ∫(f 󸀠 ∗ δ s )ds, and lim 󵄩󵄩󵄩 + 󵄩 󵄩󵄩L1ω t t→0 󵄩 0

Proof. Set R(s)g = g ∗ δ s fo s > 0, g ∈ L1ω . Then R(s) is a strongly continuous semigroup of bounded operators on L1ω , and ‖R(s)‖ ≤ ω(s) for s > 0, and so the

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integral ∫0 (f 󸀠 ∗ δ s )ds is well-defined in L1ω . Since f 󸀠 is continuous, this integral is also well-defined in L1ω ∩ C([0, L]) for every L > 0, and we have, for x ≥ 0, +∞

t

t

t

0

0

0

(∫(f 󸀠 ∗ δ s )ds) (x) =< ∫(f 󸀠 ∗ δ s )ds, δ x >= ∫ < f 󸀠 ∗ δ s , δ x > ds t

= ∫(f 󸀠 ∗ δ s )(x)ds.

We have (f 󸀠 ∗ δ s )(x) = f 󸀠 (x − s) for 0 ≤ s ≤ x, and (f 󸀠 ∗ δ s )(x − s) = 0 for s > x. Hence 0

∫(f ∗ δ s )(x)ds = t

󸀠

0

∫

min(x,t)

f 󸀠 (x − s)ds = −f(x − min(x, t)) + f(x).

0

But f(x − min(x, t)) = f(x − t) for x ≥ t, and f(x − min(x, t)) = f(0) = 0 for 0 ≤ x ≤ t. 󵄩 󵄩 t So f ∗ δ t − f = − ∫0 (f 󸀠 ∗ δ s )ds for t ≥ 0, and limt→0+ 󵄩󵄩󵄩󵄩 f∗δtt −f + f 󸀠 󵄩󵄩󵄩󵄩L1 = 0. ω

We now give the classical approach to the domain of the generator of a semigroup. Proposition 8.5. (i) Let u ∈ IT . If limt→0+ ‖ T(t)u−u − v‖ = 0 for some v ∈ UT , then t u ∈ DAT,IT , and AT,IT u = v. (ii) Let u ∈ JT . Then u ∈ DAT,JT if, and only if, there exists v ∈ JT such that 󵄩 󵄩 limt→0+ 󵄩󵄩󵄩󵄩 T(t)u−u − v󵄩󵄩󵄩󵄩J = 0, and in this situation AT,JT u = v. t T

− v‖ = 0 for some v ∈ UT , let f0 ∈ Proof. (i) If u ∈ IT , and if limt→0+ ‖ T(t)u−u t C 1 ([0, +∞)) ∩ Ω (L1ωT ) satisfiying f0 (0) = 0, f0󸀠 ∈ L1ωT . It follows from the lemma that we have in IT −ϕT (f0󸀠 )u = [ lim

t→0+

T(t)ϕT (f0 ) − ϕT (f0 ) T(t)u − u ] u = ϕT (f0 ) [ lim ] = ϕT (f0 )v, t t t→0+

and so u ∈ DAT,IT , and AT,IT u = v. (ii) The same argument shows that if u ∈ JT , and if we have 󵄩󵄩 T(t)u − u 󵄩󵄩 󵄩 󵄩 lim 󵄩󵄩󵄩 − v󵄩󵄩󵄩 = 0 󵄩󵄩JT t t→0+ 󵄩 󵄩

for some v ∈ JT , then u ∈ DAT,JT , and AT,JT u = v. Conversely assume that u ∈ DAT,JT , and let f0 ∈ C 1 ([0, +∞)) ∩ Ω (L1ω̃ T ) . Set v = AT,JT u. We have, for t ≥ 0, t

t

t

0

0

0

ϕ̃ T (f0 ) ∫ T(s)vds = ∫ T(s)ϕ̃ T (f0 )vds = − ∫ T(s)ϕ̃ T (f0󸀠 )uds

= − [∫ T(s)ϕ̃ T (f0󸀠 )ds] u = [T(t)ϕ̃ T (f0 ) − ϕ̃ T (f0 )] u = ϕ̃ T (f0 )(T(t)u − u). ] [0 t

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Since ϕ̃ T (f0 ) ∈ Ω(JT ), this shows that T(t)u − u = ∫0 T(s)vds, and so t

󵄩󵄩 󵄩󵄩 T(t)u − u 󵄩 󵄩 − v󵄩󵄩󵄩 = 0. lim 󵄩󵄩󵄩 + 󵄩󵄩JT t t→0 󵄩 󵄩

9 The Arveson spectrum

̂ Assume that IT is not radical, and denote again by I T the space of characters of IT , equipped with the usual Gelfand topology. The Arveson spectrum σ ar (S) has been defined for S ∈ QM(IT ) by the formula ̂T }, ̃ σ ar (S) = {λ = χ(S) : χ∈I

where χ̃ denotes for χ ∈ ̂ IT the unique character on QM(IT ) such that χ̃ |IT = χ, see definition 5.5. If μ is a measure on [0, +∞), the Laplace tranform of μ is defined by the usual +∞ +∞ formula L(μ)(z) = ∫0 e−zt dμ(t) when ∫0 e−Re(z)t d|μ|(t) < +∞. We have the following easy observation. Proposition 9.1. Let μ ∈ MωT . Then we have, for χ ∈ ̂ IT , +∞

̃ T,IT )). χ̃ ( ∫ T(t)dμ(t)) = L(μ)(−χ(A

(14)

̃ T,IT )) = L(μ)(−χ(A ̃ T,JT )). χ̃ ( ∫ T(t)dμ(t)) = L(μ)(−χ(A

(15)

IT , Similarly we have, for μ ∈ Mω̃ T , χ ∈ ̂ 0

+∞

0

̃ T,IT )t ̃ In particular χ(T(t)) = e−χ(A for t > 0.

̂ ̃ Proof. If χ ∈ I is continuous T , then χ̃ |AT is a character on AT , the map t → χ(T(t)) ̃ on (0, +∞) and so there exists λ ∈ C such that χ(T(t)) = e−λt for t > 0, and |e−λt | ≤ ‖T(t)‖, which shows that Re(λ) ≥ −log(ρT ). Let u ∈ Ω(IT ), and let μ ∈ MωT . We have +∞

+∞

+∞

0

0

0

χ(u)χ̃ ( ∫ T(t)dμ(t)) = χ (u ∫ T(t)dμ(t)) = χ ( ∫ T(t)udμ(t))

On the generation of Arveson weakly continuous semigroups +∞

+∞

0

0

|

83

= ∫ χ(T(t)u)dμ(t) = χ(u) ∫ e−λt dμ(t) = χ(u)L(μ)(λ),

̃ T (μ)) = L(μ)(λ). and so χ(ϕ Let f0 ∈ C 1 ([0, +∞)) ∩ Ω(L1ωT ) such that f0 (0) = 0 and f 󸀠 (0) ∈ L1ωT . We have λL(f0 )(λ) = L(f0󸀠 )(λ) = χ(ϕT (f0󸀠 )) = −χ̃ (AT,IT ϕT (f0 )) ̃ T,IT )χ(ϕT (f0 ) = −χ(A ̃ T,IT )L(f0 )(λ), = −χ(A

̃ T,IT ), which proves (14), and formula (15) follows from a similar and so λ = −χ(A ̃ T,IT )) for t > 0. argument. In particular χ(T(t)) = L(−χ(A The following consequence of proposition 9.1 pertains to folklore. Corollary 9.2. Assume that IT is not radical. ̃ T,IT ) is a homeomorphism from ̂ (i) The map χ → χ(A IT onto σ ar (AT,IT ). (ii) The set ∆ t := {λ ∈ σ ar (AT,IT ) | Re(λ) ≤ t} is compact for every t ∈ R .

Proof. Let f0 ∈ C 1 ([0, +∞)) ∩ Ω(L1ωT ) such that f0 (0) = 0 and f0󸀠 ∈ L1ωT . We have, for χ ∈ ̂ IT , ̃ T,IT ) = − χ(ϕT (f0 )) ≠ 0, and χ(A

χ(ϕT (f0󸀠 )) , χ(ϕT (f0 ))

̃ T,IT ) is continuous with respect to the Gelfand topology and so the map χ → χ(A on ̂ IT . Conversely let f ∈ L1ωT . It follows from Proposition 9.1 that we have, for χ ∈ ̂ IT , ̃ T,IT )). χ(ϕT (f)) = L(f)(−χ(A

Since the set {u = ϕT (f) : f ∈ L1ωT } is dense in IT , this shows that the map ̃ T,IT ) is one-to-one on ̂ χ → χ(A IT , and that the inverse map σ ar (AT,IT ) → ̂ IT is continuous with respect to the Gelfand topology. ̃ T,IT )) ≤ t}. Then |χ(T(1))| ̃ Now let t ∈ R , and set U t := {χ ∈ ̂ IT : Re(χ(A ≥ e−t for χ ∈ U t , and so 0 does not belong to the closure of U t with respect to the weak∗ topology on the unit ball of the dual of IT . Since ̂ IT ∪ {0} is compact with respect to this topology, U t is a compact subset of ̂ IT , and so the set ∆ t is compact.

10 The resolvent If A is a commutative Banach algebra with unit element e, we will set as usual specA (u) := {λ ∈ C | u − λe is not invertible in A},

̂denotes the sspace of characters on A. so that specA (u) = {χ(u)}χ∈Â, where A We now wish to discuss the Arveson resolvent of the generator of a semigroup

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T = (T(t))t>0 which is weakly continuous in the sense of Arveson. We identify JT , M(JT ) and AT to subalgebras of the algebra QM r (IT ) of regular quasimultipliers on the Arveson ideal IT . We also identify the algebras QM(IT ) and QM(JT ), and the algebras QM r (IT ) and QM r (JT ) by using the pseudobounded isomorphisms described in the previous sections. From now on we will write AT = AT,IT = AT,JT , and we will denote as before by DAT ,JT (resp. DAT ,IT , resp. DAT ,UT ) the domain of AT considered as a quasimultiplier on JT (resp. on IT , resp. on UT ). Definition 10.1. The Arveson resolvent set of AT is defined by the formula Res ar (AT ) = C \ σ ar (AT ), with the convention σ ar (AT ) = 0 if IT is radical.

We will denote by I the unit element of QM(IT ), so that JT ⊕ C I is a closed subalgebra of the Banach algebra M(JT ). We set as above u λ (t) = e λt for λ ∈ C , t ≥ 0. We obtain the usual "resolvent formula". Proposition 10.2. (i) The quasimultiplier λI − AT ∈ QM(IT ) admits an inverse (λI − AT )−1 ∈ JT ⊂ QM r (IT ) for λ ∈ Res ar (AT ). (ii) The map λ → (λI − AT )−1 is an holomorphic map from Res ar (AT ) into JT . Moreover we have, for Re(λ) > log(ρT ), +∞

(λI − AT )−1 = ϕ̃ T (u−λ ) = ∫ e−λs T(s)ds ∈ JT , 0

where the Bochner integral is computed with respect to the strong operator topology +∞ on M(JT ), and ‖(λI − AT )−1 ‖JT ≤ ∫0 e−Re(λ)t ‖T(t)‖dt.

Proof. Assume that Re(λ) > log(ρT ), let v ∈ JT , and set a = ϕ̃ T (u−λ ). We have +∞

+∞

+∞

0

0

0

av = ∫ e−λs T(s)vds, T(t)av − av = ∫ e−λs T(s + t)vds − ∫ e−λs T(s)vds +∞

+∞

t

t

0

0

= e λt ∫ e−λs T(s)vds − ∫ e−λs T(s)vds = (e λt − 1)av − e λt ∫ e−λs T(s)vds.

Since limt→0+ ‖T(t)v − v‖JT = 0, we obtain

󵄩󵄩 T(t)av − av 󵄩󵄩 󵄩 󵄩 lim 󵄩󵄩󵄩 − λav + v󵄩󵄩󵄩 = 0, + 󵄩󵄩JT t t→0 󵄩 󵄩

and so av ∈ DAT ,JT , and AT (av) = λav − v. This shows that aJT ⊂ D AT ,JT , and that (λI − AT )av = v for every v ∈ JT . We have λI − AT = S u/v , where u ∈ JT , v ∈ Ω(JT ), and we see that ua = v. Hence u ∈ Ω(IT ), λI − AT is invertible in QM(JT ), and

On the generation of Arveson weakly continuous semigroups

| 85

+∞ (λI − AT )−1 = a = ϕ̃ T (u−λ ) = ∫0 e−λt T(t)dt ∈ JT , where the Bochner integral is computed with respect to the strong operator topology on M(JT ). Set again a = ϕ̃ T (u−λ ) = (λI − AT )−1 , where λ is a complex number satisfying Re(λ) > log(ρT ). Let χ0 be the character on JT ⊕ C I such that Ker(χ0 ) = JT . Every character on JT ⊕ C I distinct from χ0 has the form χ̃ |JT ⊕CI for some χ ∈ ̂ IT . It follows 1 ̂ ̃ ̃ T )) = λ−χ(A for χ ∈ I , and we obtain, from formula (15) that χ(a) = L(u−λ )(−χ(A T ̃ T) for μ ∈ C ,

spec JT ⊕CI (I + (μ − λ)a) = {1} ∪ {

̃ T) μ − χ(A . } ̃ T ) χ∈I λ − χ(A ̂ T

Hence, I + (μ − λ)a is invertible in JT ⊕ C e for every μ ∈ Res ar (AT ), and the map μ → a(I + (μ − λ)a)−1 ∈ IT is holomorphic on Res ar (AT ). We have, for μ ∈ Res ar (AT ), (μI − AT )a(I + (μ − λ)a)−1 = ((μ − λ)I + (λ − AT ))a(I + (μ − λ)a)−1 = (I + (μ − λ)a)(I + (μ − λ)a)−1 = I.

Hence (μI − AT ) has an inverse (μI − AT )−1 ∈ JT for λ ∈ Res ar (AT ), and the map μ → (μI − AT )−1 = a(I + (μ − λ)a)−1 is holomorphic on Res ar (AT ).

If we consider AT as a quasimultiplier on JT , the fact that (μI − AT )−1 ∈ JT is the inverse of μI − AT for μ ∈ Res(AT ) means that (μI − AT )−1 v ∈ DAT ,JT and that (μI − AT ) ((μI − AT )−1 v) = v for every v ∈ JT , and that if w ∈ DAT ,JT , then (μI − AT )−1 ((μI − AT )w) = w. When lim supt→0+ ‖T(t)‖ = +∞, the situation is slightly more complicated if we consider AT as a quasimultiplier on IT . In this case the domain D(μI−AT )−1 ,IT of (μI − AT )−1 ∈ QM(IT ) is a proper subspace of IT containing UT ⊃ ∪t>0 T(t)IT , and we have (μI − AT )−1 v ∈ DAT ,IT and (μI − AT ) ((μI − AT )−1 v) = v

for every v ∈ D(μI−AT )−1 ,IT . Also if w ∈ DAT ,IT , then (μI − AT )w ∈ D(μI−AT )−1 ,IT , and we have (μI − AT )−1 ((μI − AT )w) = w.

In order to interpret (λI − AT )−1 as a partially defined operator on IT for Re(λ) > log(ρT ), we can use the formula +∞

(λI − AT )−1 v = ∫ e−λt T(t)vdt (v ∈ UT ),

(16)

0

which defines a quasimultiplier on IT if we apply it to some v ∈ Ω(IT ) ∩ UT . The fact that this quasimultiplier is regular is not completely obvious but follows from the previous discussion since (λI − AT )−1 ∈ JT ⊂ QM r (IT ). Notice that since ∪t>0 T(t)IT is dense in (UT , ‖.‖UT ), (λI −AT )−1 is characterized by the simpler formula +∞

(λI − AT )−1 T(s)v = e λs ∫ e−λt T(t)vdt (s > 0, v ∈ IT ). s

(17)

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Using the correspondence between quasimultipliers on IT and partially defined operators on X1 = [∪t>0 T(t)X]− and X0 = X1 / ∩t>0 Ker(T(t)), we obtain a new approach to the resolvent formula and the other results of section 4.

11 Holomorphic functional calculus For α ∈ R , set

Π α+ := {z ∈ C : Re(z) > α}, Π α := {z ∈ C : Re(z) ≥ α}. +

Denote as usual by H 1 (Π α+ ) the space of all holomorphic functions F on Π α+ such +∞ that ‖F‖1 := supt>0 ∫−∞ |F(t + iy)| dy < ∞. Then (H 1 (Π α+ ), ‖.‖1 ) is a Banach space, the formula F ∗ (α + iy) = limt→0+ F(α + iy + t) defines a.e. a function F ∗ on α + iR , +∞ ∫−∞ |F ∗ (α + iy)|dy < +∞, and we have +∞

lim ∫ |F ∗ (α + iy) − F(α + t + iy)|dy = 0.

t→0+

(18)

−∞

Moreover ‖F‖1 = ∫−∞ |F ∗ (α + iy)|dy for F ∈ H 1 (Π α+ ), and if we set +∞

+∞

f(t) :=

1 αt e ∫ F ∗ (α + iy)e iyt dy, 2π −∞

then it follows from the F. and M. Riesz theorem for the half-plane that f(t) = 0 for t < 0, and L(f)(z) = F(z) for z ∈ Π α+ . So we can define the inverse Laplace transform L −1 (F) of F by the formula L −1 (F)(t) =

+∞

1 αt e ∫ F ∗ (α + iy)e iyt dy (t ≥ 0). 2π

(19)

−∞

It follows then from the inversion formula for Fourier transforms that L −1 (F)(t) =

+∞

1 βt e ∫ F(β + iy)e iyt dy 2π −∞

for every β > α, so that supt≥0 e−βt |L(F)(t)| < +∞, and L −1 (F) ∈ L∞ (R+ , e−βt ), and the tautological formula L −1 (F) = L −1 (F|Π+ ) holds for F ∈ H 1 (Π α+ ), β > α, t ≥ β

0. All these results are well-known, see for example [24], Ch. 2. Notice that if F ∈ H 1 (Π α+ ), we have, for Re(z) ≥ β, β > α,

󵄨󵄨 󵄨󵄨 +∞ +∞ 󵄨󵄨 󵄨󵄨 1 1 ‖F‖1 󵄨 󵄨 |F(z)| = 󵄨󵄨󵄨 ∫ L −1 (F)(t)e−zt dt󵄨󵄨󵄨 ≤ ‖F‖1 ∫ e(α−β)t dt = . 󵄨󵄨 2π 󵄨󵄨 2π β−α 󵄨󵄨 󵄨󵄨 0 0

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Hence F|Π+ ∈ H ∞ (Π β+ ) for β > α, where H ∞ (Π β+ ) denotes the algebra of all β

bounded holomorphic functions on Π β+ . This shows that FG ∈ H 1 (Π γ+ ) if F ∈ H 1 (Π α+ ), G ∈ H 1 (Π β+ ), γ > β ≥ α, and L(L −1 (F) ∗ L −1 (G))(z) = F(z)G(z) for Re(z) ≥ γ, which shows that L −1 (FG) = L −1 (F) ∗ L −1 (G). So ∪α 0, the weighted group algebra ℓ1 (Z , w) is even Arens regular as observed in [16]. So when w is not totally bounded, Theorem 4.24 fails badly.

Theorem 4.26. Let G be an infinite locally compact group and w be a diagonally bounded weight on G with bound D. Then (1) the measure algebra M(G) is isometrically enAr (i.e., when w = 1). (2) the weighted measure algebra M(G, w) is r-enAr with distortion r at most D. For our next theorem and for the rest of the paper, we recall that the Fourier algebra is the collection of all functions h on G of the form f ̄ ∗ ǧ with f, g ∈ L2 (G) and norm ||h|| = inf {||f||2 ||g||2 : h = f ̄ ∗ g,̌ f, g ∈ L2 (G)} .

The Banach dual of A(G) is the group von Neumann algebra VN(G) which is the closure in the weak operator topology of the linear span of {λ(x) : x ∈ G} in B(L2 (G)), where λ is the left regular representation of G on L2 (G) (see [22]). A subspace of VN(G) which shall be of our interest is UC2 (G). This is defined by UC2 (G) = VN(G).A(G)

||.||

||.||

= {T ∈ VN(G) : suppT is compact}

,

where the support suppT of T is in the sense of [22, page 227]. Recall that VN(G).A(G) is automatically closed when G is amenable. This is usually called the space of uniformly continuous functionals on G, and is often denoted by UC(̂ G) in the literature. We are using the notation UC2 (G) to avoid possible confusion arising from the usage of the dual group ̂ G of G. The space UC2 (G) is in fact a C*-subalgebra of VN(G) as observed by Lau in [60, Proposition 4.4] and proved by Granirer in [46, Proposition 2(a)]. This is obtained by regarding UC2 (G)

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as the space generated by operators in VN(G) with compact support and using [22, Proposotion 4.8 (5o )]. Following [22], let P(G) be the space of continuous positive definite functions on G and B(G) be its the linear span. The space B(G) is a Banach algebra, called the Fourier-Stieltjes algebra, and if C∗ (G) is the group C∗ -algebra of G, then B(G) is its Banach dual. The Fourier algebra A(G) is a closed ideal of B(G) and can be seen as the norm-closed linear span of P(G) ∩ C00 (G) in B(G), where C00 (G) be the space of continuous functions with compact support. For more details on Fourier algebra and Fourier-Stieltjes algebra on locally compact groups, we direct the reader to the classical article [22] by Eymard and to the recent book [59] by Kaniuth and Lau. When G is abelian and ̂ G is its dual group, then via the Fourier transform, A(G) identifies with L1 (̂ G) , VN(G) with L∞ (̂ G), UC2 (G) with LUC(̂ G), P(G) with the set of Fourier transforms of all probability measures on ̂ G and B(G) with M(̂ G). So all the results stated for the group algebra and the measure algebra hold for the Fourier algebra and the Fourier-Stieltjes algebra, respectively, when G is abelian. However, unlike the case of L1 (G), non-Arens regularity of A(G) has turned out to be more resistant in general, and the solution is still not complete. We first look at non-discrete groups. Using TI-nets which are available in A(G) in this case, we are able to construct orthogonal ℓ1 (χ(G))-set in A(G) (the predual of VN(G)) and apply Theorem 4.13. Satement (ii) of the following theorem was proved in [69]. Theorem 4.27. Let G be a non-discrete locally compact group with compact covering κ(G) and local weight χ(G). Then VN(G) , (1) there is an isometric copy of ℓ∞ (χ(G)) in WAP(A(G) (2) in particular, A(G) is non-Arens regular, (3) A(G) is isometrically enAr when χ(G) ≥ κ(G). For discrete groups, we no longer have TI-nets in A(G). But when the group is amenable, we can replace TI-nets by weak bounded approximate identities. In fact, we can prove that for any locally compact group that contains a σ-compact, non-compact open amenable subgroup H, the Fourier algebra A(G) has a sequential orthogonal weak bai. Applying then Theorem 4.13 yields the following theorem. Theorem 4.28. Let G be a locally compact group with a σ-compact, non-compact open amenable subgroup H. Then VN(G) (1) there is an isometric copy of ℓ∞ in WAP(A(G) , (2) in particular, A(G) is non-Arens regular, (3) A(G) is isometrically enAr when G is second countable.

114 | Filali-Galindo Theorem 4.28 improves the results claimed in Subsection 4.7 and Remark 4.14, that VN(G) is non-separable (and so A(G) is not Arens regular) the quotient space WAP(A(G)) when G is discrete and contains an amenable subgroup. In particular, we can restate Corollary 4.29 as follows. Corollary 4.29. The Fourier algebra A(F r ) is isometrically enAr but not sAir for every r ≥ 2.

5 On the strong Arens irregularity of Banach algebras 5.1 What is known on strong Arens irregularity and topological centres We first collect what is known about topological centres of Banach algebras in harmonic analysis. Let G be a locally compact group. ∙ ∙

∙ ∙

∙ ∙

The group algebra is sAir for any infinite locally compact group. This was proved first by Isik, Pym and Ülger in [57] when G is compact, then in general by Lau and Losert in [63]. The topological centre of LUC(G)∗ is M(G). This was proved by Grosser and Losert in [49] when G is Abelian, then in general by Lau in [61]. A number of articles offering different approaches and various properties related to topological centres of LUC(G)∗ and L1 (G)∗∗ appeared subsequently, we cite for example [16], [17], [36], [75] and more recently [7]. The semigroup algebra is sAir when S is an infinite, discrete, weakly cancellative, right cancellative semigroup (see [36] and [61]). The measure algebra M(G) was proved to be sAir for a certain class of groups (see [77]), this class of groups comes up again in Theorem 6.3 below. Recently, Losert et al proved that M(G) is sAir for any infinite locally compact group (see [73]). The weighted group algebra with a diagonally bounded weight is sAir (see [16], [37] and [79]). The topological centre of LUC(G, w−1 )∗ is M(G, w) for any diagonally bounded weight. (see [37] and [79]).

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The weighted semigroup algebra ℓ1 (S, w) with a diagonally bounded weight w is sAir when S is an infinite, discrete, weakly cancellative, right cancellative semigroup (see [15] and [37]). The Fourier algebra A(G) turned out to be more resistant. As mentioned above, it is not yet known whether A(G) is non-Arens regular for all infinite discrete groups, let alone its strong Arens irregularity or its extreme non-Arens regularity. There is however a large class of groups for which the topological centres of A(G)∗∗ is A(G) and that of UC2 (G)∗ is B(G). This class includes:

∙ ∙ ∙ ∙ ∙ ∙ ∙

of course the abelian groups since in this case A(G) is identifies with L1 (̂ G) ̂ via the Fourier transform, where G is the dual group of G; the infinite discrete amenable groups (see [64]); the second countable amenable groups G such that the closure of the commutator of G, [G, G], is not open in G (see [64]); the groups of the form ∏∞ i=0 G i where each G i , i ≥ 1, is a non-trivial metrizable compact group and G0 is a second countable locally compact amenable group (see [65]); the groups of the form G × H, where G is a compact group whose local weight χ(G) has uncountable cofinality and H is any locally compact amenable group with χ(H) ≤ χ(G) (see [32]); and the compact group SU(2) (as claimed in [72]). Most of these and other algebras are shown recently in [31] to be sAir. This will be the matter of this section.

Remark 5.1. The groups for which A(G) is known to be neither Arens regular nor sAir (i.e., A(G) ⊊ T1 Z(A(G)∗∗ ) ⊊ A(G)∗∗ ) have been recently found by Losert. These include the discrete groups containing F r , the free group with r generators, where r ≥ 2 is finite (see [71]) ; the compact group SU(3) and the locally compact group SL(2, R) (see [72]). Conjecture 5.2. In the survey [62], among the interesting problems Tony Lau listed, was the determination of the topological centre of A(G)∗∗ . Ten years later, the same problem appeared again in the survey [58] by Eberhard Kaniuth and Tony Lau. Some advance has been made since then but the following conjecture is still open: The Fourier algebra is strongly Arens irregular whenever G is amenable as a discrete group.

116 | Filali-Galindo

5.2 The new approach: Fℓ1 (η)-bases and factorizations We start by defining some special types of ℓ1 (η)-bases in A, which we shall call Fℓ1 (η)-bases (the letter F stands for factorization), where η is an infinite cardinal. Theorems 5.10 and 5.11 show how Fℓ1 (η)-bases in A provide η-factorizations in A∗ , LUC(A), and in some cases also in M(A)∗ . If the Mazur property happens to be also of level η, then this yields the topological centres. Theses new results are taken from [31]. Next definition was used in [32]; it is a generalization of the factorization formula used in [76, Lemma 2.2], where m was equal to 1. Definition 5.3. Let A be a Banach algebra and η be a cardinal number. Then the Banach dual A∗ of A (respectively, LUC(A)) is said to have the η-factorization property if for every bounded family of functionals {S λ : λ < η} contained in A∗ (respectively, in LUC(A)), there exist a bounded family {ψ λk : λ < η, k = 1, ..., m} in the unit ball of LUC(A)∗ and a finite family of functionals {T k : n = 1, ..., m} in A∗ (respectively, in LUC(A)) such that the factorization formula S λ = ∑ ψ λk ⋅ T k m

k=1

holds for all

λ < η.

(1)

If the elements {ψ λk : λ < η, k = 1, ..., m} can be chosen in LUC(A)∗ independently of the elements S λ , then we say that A has the uniform η-factorization property. (This is going to be so with all our cases.) In order to apply the factorization Theorem 5.5 below to the problems of topological centre of A∗∗ and LUC(A)∗ we recall the following definition ([77]). Definition 5.4. Let E be a Banach space and η an infinite cardinal number. (1) A functional f ∈ E∗∗ is called η − w∗ -continuous if for every infinite net (x α )α in the unit ball of E∗ of cardinality at most η with x α → 0 in the w∗ -topology, we have f(x α ) → 0. (2) We say that E has the η-Mazur property if every η − w∗ -continuous functional f ∈ E∗∗ is an element of E. Next theorem is a generalization of two results by Neufang (see [77, Theorem 2.3 and Theorem 4.2]), where the factorization property is replaced with the ηfactorization property as in Definition 5.3. The proof is a straightforward modification of the original proof. Theorem 5.5. Let A be a Banach algebra and η be an infinite cardinal. If A has the η-Mazur property and its dual A∗ has the η-factorization property, then (1) A is left strongly Arens irregular, that is, T1 Z(A∗∗ ) = A;

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(2) if A has a brai bounded by 1, then the T1 Z(LUC(A)∗ ) is isometrically isomorphic to the multiplier algebra M(A) of A, that is, T1 Z(LUC(A)∗ ) ≅ M(A).

5.3 Fℓ1 (η)-bases The elements ψ λn used in (1) in Definition 5.3 for the factorization of operators in A∗ and LUC(A) are taken from the weak∗ -closure of some especially constructed sets in the multiplier algebra M(A). These sets together with an increasing family of sets in A generate the required Fℓ1 (η)-bases in A. To handle the different algebras under study, three types of Fℓ1 (η)-bases are needed. The work related to the results in the rest of the survey is still in progress in [31]. Definition 5.6. Fℓ1 (η)-base of type 1: In a Banach algebra A, an Fℓ1 (η)-base of type 1 is a set of the form L = ⋃α 0 and constant K δ > 0. On the other hand, it follows by action of Θ in Lemma 2.5 that t n−1 d φ(t; x, ℓ) = φ(t; Ax, ℓ) + ⟨x, ℓ⟩, dt (n − 1)!

t > 0,

where ℓ ∈ Y ⊥ , x = Θ(f)y, y ∈ X, f ∈ D(0, ∞) and Ax := Θ(−f 󸀠 )y. Identifying the coefficients of the corresponding power series in the above equality one then finds C k+1 (x, ℓ) = C k (Ax, ℓ) for k ≥ n. For ̃ A[x] := Ax, with x as above we have ⟨̃ A[x], ℓ⟩ = ⟨Ax, ℓ⟩ = C n+1 (x, ℓ).

(n) By (8) and the density of D(0, ∞) in T+ (t n ), ̃ A is a bounded operator on X/Y. Also, by induction one proves that C k (x, ℓ) = ⟨̃ A k−n x, ℓ⟩ for all k ≥ n. Hence, ∞ tk φ(t; [x], ℓ) = ∑ ⟨̃ A k−n [x], ℓ⟩ , t ≥ 0, k! k=n

for all [x] ∈ X/Y. It follows that

t ̃ An ) ⟨e t A [x], ℓ⟩ = ∑ ⟨̃ A k [x], ℓ⟩ + φ(t; [x], ℓ ∘ ̃ k! k=0 n−1

k

(note that the composition ℓ ∘ ̃ A n lies in (X/Y)∗ = Y ⊥ ). Thus, using (2) for x ∈ X and ℓ ∈ Y ⊥ , ̃

|⟨e t A [x], ℓ⟩| ≤ K n ‖x‖ ‖ℓ‖ |t|n ,

t ∈ R,

where K n is a constant only depending on n. Then a straightforward application ̃ of the Banach-Steinhaus principle implies that the (holomorphic) group (e t A )t∈R in B(X/Y) has polynomial growth in t ∈ R. Step 4. Spectral argument. As a matter of fact, if f ∈ D(0, ∞), λ > 0 and ̃ λ )Θ(f ̃ 󸀠) = gλ (u) := e−λu (u > 0), we have gλ ∗ f 󸀠 = f − λgλ ∗ f which implies that Θ(g ̃ ̃ λ )Θ(f); ̃ ̃ λ )Θ(f) ̃ = Θ(f). ̃ Θ(f)− λ Θ(g that is, (λ − ̃ A)Θ(g Putting a bounded approximate

148 |

Galé, Martínez and Miana

(n) ̃ identity for T+ (t n ) in the place of f , and passsing to limits we obtain Θ(g) = (I − ̃ A)−1 , for λ = 1 in particular. Then, for z ∈ C,

̃ I − (z + 1)Θ(g) = I − (z + 1)(I − ̃ A)−1 = (−z + ̃ A)(I − ̃ A)−1 .

̃ In conclusion, (z + 1)−1 lies in the spectrum of Θ(g) in B(X/Y) if and only if z ̃ ̃ ̃ belongs to the spectrum σ(A) of A in B(X/Y). Thus σ(A) is at most countable (and compact). Now, we finish using an argument similar to [17, p. 236]: Suppose ℓ ≠ 0, if ̃ possible. Since the group e t A has polynomial growth on R it follows that σ(̃ A) is ̃ nonempty in iR, by [17, Lemma 5]. Hence there is an isolated point iω in σ(A). Using the projection associated with iω by the holomorphic functional calculus one gets some nonzero ϕ ∈ X ∗ such that ̃ A∗ ϕ = iωϕ. The equality says in particular ̃ ̃ that ⟨ϕ, ̃ A Θ(g)x⟩ = iω⟨ϕ, Θ(g)x⟩ for every x ∈ X. ̃ On the other hand, from (I − ̃ A)Θ(g) = I it follows that that is,

̃ ̃ ̃ ⟨ϕ, x⟩ = ⟨ϕ, Θ(g)x⟩ − ⟨ϕ, ̃ A Θ(g)x⟩ = ⟨ϕ, Θ(g)x⟩ − iω⟨ϕ, x⟩; ̃ ⟨ϕ, Θ(g)x⟩ = (1 − iω)⟨ϕ, x⟩ = L(g)(−iω)⟨ϕ, x⟩.

Since g is a polynomial generator of the Sobolev algebra, by continuity it follows ̃ that ⟨ϕ, Θ(f)x⟩ = L(f)(−iω)⟨ϕ, x⟩ for every x ∈ X. In other words, we obtain ∗ ̃ ̃ ∗ ) ∩ iR, but this set is empty Θ (f)ϕ = L(f)(−iω)ϕ. This means that −iω is in Pσ(Θ by hypothesis. Thus we have arrived at a contradiction, so that ℓ must be zero and then we are done. One would certainly like to remove that condition of being an interpolation set, as well as the compactness, on the subset iσ(Θ) ∩ R in Proposition 4.1. In any case, the following question is in order. (n)

Question 4.2. What real subsets are interpolation sets for T+ (t n ) in T (n) (|t|n ) ? A consequence of Lemma 2.2 is that finite sets in R are of interpolation. This implies the result which follows. (n)

Corollary 4.3. Let Θ : T+ → B(X) be a Banach algebra bounded homomorphism (n) with Θ(T+ )X dense in X. Assume that σ(Θ) ∩ iR is finite and Pσ(Θ∗ ) ∩ iR = 0.

Then Θ n (M n,+ (S))X is dense in X. Remark 4.4. The problem approached in this section was suggested by work done in [7], connected with the asymptotic behavior of bounded C0 -semigroups on a

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149

Banach space X. Let T0 (t) = e tA be such a semigroup, with infinitesimal generator A. The Arendt-Batty-Lyubich-Vũ stability theorem says that limt→∞ T0 (t)x = 0 for all x ∈ X if σ(A)∩iR is countable and σ P (A∗ )∩iR = 0, see [1] and [13]. (σ P (A∗ ) is the point spectrum of A∗ .) The theorem has been extended or generalized in so many directions. For example in [17], for semigroups with nonquasianalytic growth. On the other hand, Esterle-Strousse-Zouakia in [7] and Vũ in [16] established the following continuous version of the Katznelson-Tzafriri theorem: For T0 (t) and A as above, let π0 : L1 (R+ ) → B(X) be the bounded homomorphism given for ∞ f ∈ L1 (R+ ) and x ∈ X by π0 (f)x := ∫0 f(t)T0 (t)x dt. Then ‖ limt→∞ T0 (t)π0 (f)‖ = 0, for all f ∈ M0,+ (S), where S := iσ(A) ∩ R. In [7, Théorème 3.7], it is also proven that π0 (M0,+ (S))X is dense in X, provided that σ(A)∩iR is a countable set and σ P (A∗ )∩ iR = 0. This shows us a new proof of the Arendt-Batty-Lyubich-Vũ theorem, as it is noticed in [7, p. 215]. It is not clear what should be the most accurate version of the Arendt-BattyLyubich-Vũ theorem in the setting of n-times integrated semigroups (for definitions and part of the theory of integrated semigroups one can see [3, Section 3.2]; every C0 -semigroup is a 0-integrated semigroup). In [6, Theorem 5.6], it is shown that for a bounded once integrated semigroup (T1 (t))t≥0 with invertible generator A such that σ(A) ∩ iR is countable and σ P (A∗ ) ∩ iR = 0 one has lim T1 (t)x = −A−1 x

t→∞

(x ∈ D(A)).

Let (T n (t))t≥0 be a n-times integrated semigroup with not necessary invertible generator A, satisfying supt>0 t−n ‖T n (t)‖ < ∞ and span(T n (t)X) dense in X. Under these assumptions it has been shown in [8] that, in B(X), one has lim t−n T n (t)π n (f) = 0 (f ∈ M n,+ (S)),

t→∞

where S := iσ(A)∩ R, and π n (f)x := (−1)n ∫0 f (n) (t)T n (t)x dt (x ∈ X). Hence, if one can prove that π n (M n,+ (S))X is dense in X then one obtains limt→∞ t−n T n (t)x = 0 for all x ∈ X. It turns out that the discussion carried out in Section 4 admits an exposition in terms of integrated semigroups, where π n is to play the role of the morphism Θ n and T n (t) = Θ n (R n−1 ). In this way one applies Corollary 4.3 (and similarly with t Proposition 4.1 if wished) to get ∞

Corollary 4.5. Let T n (t) be a n-times integrated semigroup satisfying T n (t)X dense in X and supt>0 t−n ‖T n (t)‖ < ∞, and with generator A such that iσ(A) ∩ R is finite and σ P (A∗ ) ∩ iR = 0. Then lim t−n T n (t)x = 0,

t→∞

x ∈ X.

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Despite the interest of the n-integrated semigroups, we have thought that the spectral synthesis is interesting in connection with morphisms in its own, and so have chosen to present the matter as formerly.

Bibliography [1] [2]

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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A. Ya. Helemskii

Projective and free matricially normed spaces Abstract: We introduce and study metrically projective and metrically free matricially normed spaces. We describe these spaces in terms of a special space ̂ Mn , the space of n × n matrices, endowed with a special matrix–norm. We show that metrically free matricially normed spaces are matricial ℓ1 –sums of some distinguished families of matricially normed spaces ̂ M n , whereas metrically projective matricially normed spaces are complete direct summands of matricial ℓ1 –sums of arbitrary families of spaces ̂ M n . At the end we specify the underlying normed ̂ space of M n and show that the spaces ̂ M n do not belong to any of the classes L p ; p ∈ [1, ∞], introduced by Effros and Ruan. However, in a certain sense the behavior of ̂ M n resembles that of L1 –spaces. Keywords: Proto-quantum spaces, quantum spaces, proto-quantum L p –spaces, proto-operator-projective tensor product. Classification: 46L07, 46M05.

1 Introduction The notion of a matricially normed space was introduced by Effros and Ruan [3]. Very soon, after the discovery of Ruan Representation Theorem [15], the most attention was attracted by the outstanding special class of these structures, the L∞ – spaces, more often called now (abstract) operator spaces. (see the textbooks [2, 4, 12, 14]). However, already in [3, 15] it was demonstrated that matricially normed spaces are a subject of considerable interest also outside the class of operator spaces. In particular, according to these papers, one can successfully study the Haagerup tensor product of these spaces. Later it was shown in [10] that another important tensor product of operator spaces, the projective tensor product, also can be studied in the context of general matricially normed spaces. This research was supported by the Russian Foundation for Basic Research (grant No. 15-0108392). A. Ya. Helemskii , Moscow State (Lomonosov) University, Moscow, 111991, Russia, e-mail: [email protected] https://doi.org/10.1515/9783110602418-0008

152 | Helemskii In this paper we introduce and study the closely connected notions of a metrically projective and a metrically free matricially normed space. In the realm of operator spaces the definition of a metrically projective space resembles what was called by Blecher [1] (just) projective space, but differs from the latter. In the classical context of Banach spaces the metric projectivity appeared, under different name, in the old paper of Graven [5]. We characterize metrically free matricially normed spaces and then, using the well known general categorical connection between the freeness and the projectivity, metrically projective matricially normed spaces. We describe the spaces in both classes in terms of the special space ̂ M n , the space of n × n–matrices endowed with a special matrix–norm. We show that metrically free spaces are matricial ℓ1 -sums of some specified families of spaces ̂ M n whereas metrically projective spaces are complete direct summands of matricial ℓ1 -sums of arbitrary families of spaces ̂ Mn . The latter result, concerning projectivity, resembles Blecher’s Theorem 3.10 in [1], and in fact it was inspired by that theorem. Note that the mentioned theorem, extended in a straightforward way to general matricially normed spaces, also can be deduced from the description of metrically free matricially normed spaces, however after some elaboration of our general categorical tools; cf. [8]. But we leave this material outside the scope of the present paper. The contents of the paper is as follows. The second section contains some initial definitions. In Section 3 we introduce our main matricially normed space ̂ Mn . In Section 4 we prepare our tools from category theory. We consider the socalled rigged categories (well known under many different names), define projective objects in such a category and show that the metric projectivity of matricially normed spaces is a particular case of this general categorical projectivity. Then we introduce, within the frame–work of a rigged category, the notion of a free object. We recall several general categorical observations that will be used in later sections, notably the characterization of projective objects as retracts of free objects. In Section 5 we fix n ∈ N and introduce the special rig ‘⊙n ’, playing, in a sense, the role of a ‘building brick’ for the rig, responsible for the metric projectivity. We show that the ⊙n –free object with a one-point base is exactly the matricially normed space ̂ M n . The proof heavily relies on properties of a distinguished 2 2 n × n –matrix I. The latter, in the guise of an element of M n ⊗ M n , is ∑ij e ji ⊗ e ij , where e ij denotes the elementary matrix with 1 on the ij–th place. In Section 6 we apply the results of the previous sections to obtain our main results, namely the above–mentioned description of metrically free, and, as a corollary, of metrically projective matricially normed spaces.

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Finally, in Section 7 we obtain some (far from complete) information about the structure of the space ̂ M n . First, we find its underlying normed space: it turns out to be the space of n × n–matrices with the trace class norm. Thus, it is the same as the underlying space of the operator space T n , playing the main role in the description of projective operator spaces in [1]. However, as a matricially normed space, ̂ M n is profoundly different from T n . We show that it does not belong to any of the classes L p ; 1 < p ≤ ∞ of Effros/Ruan, which is not surprising, and (what is somehow surprising to the author) to the class L1 as well. Nevertheless, in a certain sense the behavior of ̂ M n resembles that of L1 –spaces.

2 Initial definitions In what follows, we denote the space of n × n-matrices, as a pure algebraic object, by M n , and the same space, endowed by the operator norm ||⋅||o or the trace norm ||⋅||t , by M n and T n , respectively. If E is a normed space, we denote by B E its closed unit ball. The identity operator on a linear space E is denoted by 1E . Let E be a linear space, M n (E) the space of n × n-matrices with entries from E. We identify M n (E) with the tensor product M n ⊗ E. According to our convenience, we shall use either ‘matrix guise’ or ‘tensor guise’ of this space. We denote the E–valued diagonal block–matrix with matrices u1 , ..., u k on the diagonal by u1 ⊕ ... ⊕ u k . Definition 2.1. (Effros/Ruan [3]) A sequence of norms || ⋅ ||n on M n (E) : n ∈ N is called a matrix–norm on E, if it satisfies two following conditions: Axiom 1. For u ∈ M n (E) and n, m ∈ N we have ||u ⊕ 0||n+m = ||u||n (Here 0 is the zero matrix in M m (E)). Axiom 2. For u ∈ M n (E) and S ∈ M n we have ||Su||n ≤ ||S||o and ||uS||n ≤ ||u||||S||o . A space E, endowed with a matrix–norm, is called matricially normed space. The normed space, identified with M1 (E), is called underlying normed space of our matricially normed space. Two examples that we shall need are the matricially normed spaces Cmin and Cmax . The first one is C with the matrix-norm, arising after the identifying, for every m ∈ N, of M m (C) with M m , whereas the second one is C with the matrix-norm, arising after the identifying of M m (C) with T m .

154 | Helemskii Every subspace F of an matricially normed space is, of course, a matricially normed space with respect of induced norms in M n (F) ⊆ M n (E) for every n. We call it matricially normed subspace of E. Now let E and F be linear spaces, φ : E → F a linear operator. The operator φ n : M n (E) → M n (F) : (x ij ) 󳨃→ (φ(x ij )), is called the n–th amplification of φ. (In the ‘tensor approach’ φ n is, of course, 1M n ⊗ φ). If our E and F are matricially normed spaces, then we call φ completely bounded, if sup{||φ n ||; n ∈ N} < ∞. We denote this supremum by ||φ||cb . If, in the previous context, every amplification is a contractive operator (that is ||φ||cb ≤ 1), we say that φ is completely contractive. (This is the most important class of operators in the present paper). The set of completely contractive operators between E and F will be denoted by CC(E, F). If every amplification is isometric, strictly coisometric or isometric isomorphism, we say that φ is completely isometric, completely strictly coisometric or completely isometric isomorphism, respectively. Here we recall that the operator between normed spaces E and F is called strictly coisometric (or exact quotient map), if it maps B E onto B F . The (non–additive) category with matricially normed spaces as objects and completely contractive operators as morphisms is denoted by MN1 . Evidently, isomorphisms in this category are completely isometric isomorphisms, defined above. Definition 2.2. A matricially normed space P is called metrically projective, if, for every completely strictly coisometric operator τ between matricially normed spaces E and F and every completely contractive operator φ : P → F there exists a completely contractive operator ψ : P → E, making the diagram P

φ

E ψ

τ

F commutative.

3 Construction of the matricially normed spaces ̂n M From now on, up to a special announcement, we fix some n ∈ N. Sometimes, if there is no danger of confusion, we omit index n. We begin with pure algebraic preparations.

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Suppose we are given a linear space E. Let us introduce the operator ι E : M n (E) → L(M n , E),

(1)

where L(⋅, ⋅) is the symbol of the space of linear operators. It takes an E–valued n× n–matrix (x ij ) to the operator (λ ij ) 󳨃→ ∑ij λ ji x ij ∈ E, where (λ ij ) ∈ M n . Equivalently, if we use the ‘tensor guise’ of M n (E), then ι E : M n ⊗ E → L(M n , E) is well defined by taking an elementary tensor a ⊗ x to the operator b 󳨃→ tr(ab)x. (Here and onwards tr denotes the trace of a matrix). Obviously, ι E is a linear isomorphism. It is convenient to denote, for a given v ∈ M n (E), the operator ι E (v) by φ v : M n → E. Now consider all possible couples (E, v), where E is a matricially normed space and v ∈ B M n (E) . In what follows, we refer them as proper couples. Definition 3.1. For every m ∈ N and u ∈ M m (M n ) we set ||u||m := sup{||(φ v )m (u)||m },

(2)

where supremum is taken over all proper couples. (We recall that (φ v )m : M m (M n ) → M m (E) takes an M n –valued m×m–matrix (a kl ) to the E–valued m × m–matrix (φ v (a kl )). Proposition 3.2. The indicated supremum is finite. Moreover, ||u||m does not exceed the sum of modules of matrix entries after the identification of M m (M n ) with M mn . Proof. Take u ∈ M m (M n ); u = (a kl ), a kl = (λ kl ij ) ∈ M n . We must show that for every proper couple (E, v) we have ||(φ v )m (u)|| ≤ ∑kl ∑ij |λ kl ij |. If v = (x ij ); x ij ∈ E, then, since v ∈ B M n (E) , and E is a matricially normed space, we have ||x ij || ≤ 1 for all i, j. Hence ||φ v (a kl )|| ≤ ∑ij |λ kl ij |. On the other hand, using again that E is a matricially normed space, we have ||(φ v )m (u)|| ≤ ∑kl ||φ v (a kl )||.

Theorem 3.3. The sequence of functions u 󳨃→ ||u||m ; m ∈ N is a matrix-norm on Mn .

Proof. First, we show that the function u 󳨃→ ||u||m is a seminorm on M m (M n ) and then we check the Axioms 1 and 2. All three assertions are proved by a similar argument; namely, we use the respective properties of spaces E in proper couples and then the definition of ||u||m as the relevant supremum. For example, the first estimation in Axiom 2 follows from the relations ||(φ v )m (Su)|| = ||S(φ v )m (u)|| ≤ ||S||o ||(φ v )m (u)|| ≤ ||S||o ||||u||m .

Finally, we prove that our seminorm is actually a norm. Take u ∈ M m (M n ); u = k󸀠 l󸀠 󸀠 󸀠 󸀠 󸀠 (a kl ); a kl = (λ kl ij ) ∈ M n . If u ≠ 0, then λ j󸀠 i󸀠 ≠ 0 for a certain k , l and i , j . Choose a

156 | Helemskii proper couple with v = (x ij ) ∈ M n (E) such that x ij ≠ 0 if, and only if i = i󸀠 and j = j󸀠 , and also ||x i󸀠 j󸀠 || ≤ 1. Then we see that (φ v )m takes u to the matrix (y kl ) ∈ M m (E) with y k󸀠 l󸀠 ≠ 0. Therefore ||(φ v )m (u)|| > 0, hence ||u|| > 0.

We denote the resulting matricially normed space by ̂ Mn . ̂ In particular, it is easy to show that M1 is just Cmax . The structure of ̂ M n for bigger n is not so transparent; we shall see this in our last section.

4 Projectivity and freeness in rigged categories Definition 4.1. Let K be an arbitrary category. A rig of K is a faithful covariant functor ◻ : K → L, where L is another category. A pair (K, ◻), consisting of a category and its rig, is called rigged category. We call a morphism τ in K admissible, if ◻(τ) is a retraction in L.

Definition 4.2. An object P in K is called ◻-projective, if, for every ◻–admissible morphism τ : E → F and every morphism φ : P → F, there exists a morphism ψ : P → F, making the diagram (D1) commutative.

Our principal example. Consider the covariant functor ⊙ : MN1 → Set,

(3)

taking a matricially normed space E to the cartesian product X∞ m=1 B M m (E) . Thus, the elements of the set ⊙(E) are sequences (w1 , . . . , w m , . . . ) where w m ∈ B M m (E) . As to the action of our functor on morphisms, it takes a completely contractive operator φ : E → F to the map ⊙(φ) : ⊙(E) → ⊙(F), (w1 , . . . , w m , . . . ) 󳨃→ (φ(w1 ), . . . , φ(w m ), . . . ). It is clear that we have obtained a rigged category. Note the following obvious statement. Proposition 4.3. (i) A morphism in MN1 is ⊙–admissible if, and only if it is a completely strictly coisometric operator. (ii) ⊙-projective matricially normed spaces are exactly metrically projective matricially normed spaces. Return to general rigged categories. The following concept is well known under different names.

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Definition 4.4. Let M be an object in L. An object Fr(M) in K is called ◻-free object with the base M, if, for every X ∈ K, there exists a bijection IX : hL (M, ◻X) → hK (Fr(M), X),

(4)

between the respective sets of morphisms, natural on X. We say that a rigged category admits freeness, if every object in L is a base of a free object in K. Remark 4.5. According to [11, Chs. III,IV], to say that a rigged category admits freeness is equivalent to say that ◻ has a left adjoint functor. The following observations show the practical use of the freeness. They are actually well known and can be extricated, as particular cases or easy corollaries, from some general facts, contained in [11, Chs. III,IV]. Proposition 4.6. Suppose that our rigged category admits freeness. Then (i) every object in K is the range of an admissible epimorphism with a free domain. (ii) an object in K is projective if, and only if it is a retract of a free object. In particular, all free objects in K are projective. It was proved in [8] that the rig, obtained from ⊙ by the restriction of MN1 to its subcategory of operator spaces, admits freeness, and its free objects are the so– called ⊕1 –sums of certain families of ‘building bricks’. The latter are the spaces T n , considered, as operator spaces, as dual to the ‘concrete’ operator space M n . (These spaces were already used in [1]). A similar result was obtained in [9] in more general context of operator modules over operator algebras. But our aim is to find free objects in the ‘whole’ rigged category (MN1 , ⊙). To begin with, we shall find free objects in another rigged category that is much simpler than (MN1 , ⊙).

5 The rig ⊙n and its free objects with the one-point base With n still fixed, we consider the covariant functor ⊙n : MN1 → Set,

(5)

taking a matricially normed space to the set B M n (E) and taking a completely contractive operator to its restriction to the respective unit balls. Evidently, we get a rig. Now we need some preparation. First, recall the operator ι E (see (1)).

158 | Helemskii Proposition 5.1. Let E, F be linear spaces, ψ : E → F an operator. Then the diagram M n (E)

ιE

L(M n , E) hψ

ψn

M n (F)

ιF

L(M n , F),

where h ψ takes an operator χ to the composition ψχ, is commutative. Proof. A convenient way to check this is to use the ‘tensor guise’ of M n (E) and look at elementary tensors in M n ⊗ E.

Recall what is e ij ∈ M n and consider the matrix I, which is ∑ij e ji ⊗ e ij after the identification of M n (M n ) with M n ⊗ M n (cf. Section 1). A routine calculation gives

Proposition 5.2. The operator φI is just 1M n . Proposition 5.3. For every linear space E and v ∈ M n (E) we have φ vn ( I) = v.

Proof. Represent v as ∑ e ij ⊗ x ij ; x ij ∈ E. Then we have φ v (e ij ) = x ji , hence (φ v )n ( I) = ∑ij e ji ⊗ φ v (e ij ) = ∑ij e ji ⊗ x ji = v.

This, together with (2), implies

Proposition 5.4. The norm of I in M n (̂ M n ) is 1.

Theorem 5.5. The matricially normed space ̂ M n is a ⊙n –free object with the onepoint set as its base. Proof. Let E be a matricially normed space, and {⋆} a one-point set. According to (4), we must construct a bijection IEn : hSet ({⋆}, ⊙n (E)) → hMN1 (̂ M n , E),

or, equivalently, a bijection IEn : B M n (E) → CC(̂ M n , E), natural on E. Take v ∈ B M n (E) and consider φ v as an operator between the matricially normed spaces ̂ Mn and E. Then for every m and u ∈ M m (̂ M n ) we have, by (2), that ||u||m ≥ ||(φ v )m (u)||. This means that φ v : ̂ M n → E is completely contractive. Thus, the map ι E has the well defined restriction to B M n (E) and CC(̂ M n , E). It is this restriction that we choose as IEn . Show that it has required properties. The commutative diagram (D2), being restricted to the respective unit balls and sets CC(⋅, ⋅), demonstrates that our constructed IEn is natural on E. Also IEn is obviously injective. It remains to show that it is surjective.

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Consider, for ψ : CC(̂ M n , E), the diagram B M n (M ̂n)

În

Mn

hψ

ψn

B M n (E)

CC(̂ Mn , ̂ Mn )

IEn

CC(̂ M n , E),

the relevant restriction of the diagram (D2) after choosing ̂ M n as E and E as F. Recall the element I; by Proposition 5.4, it belongs to B M n (M ̂ n ) . Further, Proposition 5.2 implies that În (I) = 1M ̂ n . But h ψ (1M ̂ n ) = ψ, and our diagram is commutative. Mn n Therefore ψ = IE (ψ n (I)), and we are done. .

6 Characterization of free and projective spaces To move from the rig ⊙n and its free objects with one-point bases to the ‘whole’ ⊙ and its free objects with arbitrary bases, we need the following well known categorical concept (cf., e.g., [7, Ch.2] or [11]). Let X ν ; ν ∈ Λ be a family of objects in an (arbitrary) category K. We recall that a pair (X, {i ν ; ν ∈ Λ}), where X is an object, and i ν : X ν → X are morphisms in K, is said to be the coproduct of this family, if, for every object Y and a family of morphisms ψ ν : X ν → Y there is a unique morphism ψ : X → Y such that we have ψi ν = ψ ν for every ν ∈ Λ. (We speak about ‘the’ coproduct because it is unique up to a categorical isomorphism, compatible with the respective coproduct injections.) The mentioned X, denoted in the detailed form by ∐{X ν ; ν ∈ Λ}, is referred as the coproduct object, and i ν as the coproduct injections. The morphism ψ is called the coproduct of the morphisms ψ ν and denoted by ∐{ψ ν ; ν ∈ Λ}. We say that K admits coproducts, if every family of its objects has the coproduct. Of course, the category Set admits coproducts: the coproduct of a family of sets is their disjoint union, with obvious coproduct injections. Also it is well known that the category Nor1 of normed spaces and contractive operators also admits coproducts: the coproduct of a family of normed spaces is their (classical) ℓ1 –sum. Now suppose we have a family E ν ; ν ∈ Λ of matricially normed spaces Consider their algebraic sum E := ⊕{E ν ; ν ∈ Λ} and identify, for every m ∈ N = 1, 2, ..., the linear spaces M m (E) and ⊕{M m (E ν ); ν ∈ Λ}. Endow every M m (E) with

160 | Helemskii the norm of the ℓ1 –sum of normed spaces. Then we easily see that we made E a matricially normed space. We call it matricial ℓ1 -sum of a given family. As an easy corollary of the structure of coproducts in Nor1 , we obtain Proposition 6.1. The matricial ℓ1 -sum of a given family of matricially normed spaces is the coproduct of this family in MN1 with the natural embeddings i ν : E ν → E as the coproduct injections. Thus, the category MN1 admits coproducts. Remark 6.2. Blecher [1, Sect. 3] with the help of another, necessarily more sophisticated construction, has shown that the full subcategory of MN1 , consisting of operator spaces, also admits coproducts. We turn to ⊙–free objects that in what follows will be referred as metrically free matricially normed spaces. At first we concentrate on the case of the one–point base. From now on we ‘release’ n. Denote by ̂ M∞ the matricial ℓ1 –sum ( = coproduct in MN1 ) of the family {̂ M n ; n ∈ N}.

Theorem 6.3. The metrically free matricially normed space with the one–point base, say {⋆}, does exist, and it is ̂ M∞ .

Proof. Let E be an arbitrary matricially normed space. We must construct the bijection IE∞ : hSet ({⋆}, ⊙(E)) → hMN1 (̂ M∞ , E), (6)

natural on E. The first of the indicated sets can be identified with the set of sequences w = (w1 , ..., w n , ...); w n ∈ B M n (E) . By Theorem 5.5, for every n, after relevant identifications, there exists a bijection IEn : B M n (E) → CC(̂ M n , E), taking w n to the operator φ w n . Thus every sequence w gives rise to a family of completely contractive operators φ w n : ̂ M n → E. n Denote by φw : ̂ M∞ → E the coproduct of these φ w . Taking every map from {⋆} into ⊙(E), identified with the respective w, to φw , we obtain a map IE∞ between the sets, indicated in (6). Every IEn is natural on E, and for all completely contractive operators ψ : E → F, where E, F are matricially normed spaces, we obviously have ψ(∐{φ v n }) = ∐{ψφ v n }. This easily implies that IE∞ is natural on E. Finally, since IEn is a bijection for every n, IE∞ is also a bijection. To pass from the one–point set, as the base of free objects, to arbitrary sets, we can use the following simple categorical observation. Let ◻ : K → L be an arbitrary rig. Proposition 6.4. Suppose that we are given a family F ν ; ν ∈ Λ of free objects with bases M ν . Further, suppose that there exist the coproducts F := ∐{F ν ; ν ∈ Λ} and

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M := ∐{M ν ; ν ∈ Λ} in K and L, respectively. Then F is a free object with the base M. Proof. See, e.g., [8, Prop. 2.13]. Since every set M is the coproduct of its one-point subsets, this proposition immediately implies Theorem 6.5. For every set M, there exists a metrically free matricially normed space with the base M, and it is the matricial ℓ1 -sum of the family of copies of the matricially normed space ̂ M∞ , indexed by points of M. Thus, the rigged category (MN1 , ⊙) admits freeness. Now we want to pass from free to projective matricially normed spaces. To make the formulation more geometrically transparent, we say that a matricially normed space F is a a complete direct summand of a matricially normed space E, if F is completely isometrically isomorphic to a matricially normed subspace G of E such that there is a completely contractive projection of E onto G. We have an obvious Proposition 6.6. A matricially normed space F is a retract in MN1 of a matricially normed space E if, and only if F is a complete direct summand of E. In what follows, we use a simple general-categorical observation, concerning an arbitrary rig. Proposition 6.7. (i) A retract of a ◻–projective object is ◻–projective. (ii) The coproduct of a family of ◻–projective objects (if, of course, it does exist) is ◻–projective.

We call a matricially normed space ̂ M–composed, if it is a matricial ℓ1 –sum of some family of spaces such that each of summands is ̂ M n for some n ∈ N. Theorem 6.8. (i) Every matricially normed space is an image of a completely strictly coisometric operator with the ̂ M–composed space as its domain. (ii) A matricially normed space is metrically projective if, and only if it is a complete direct summand of a ̂ M–composed space.

Proof. Combining Propositions 4.3(i) and 4.6(i) with Theorem 6.5 and Proposition 4.3, we obtain (i). Combining Propositions 4.6(ii) and 6.6 with Theorem 6.5, we obtain the ‘only if’ part of (ii). To prove the rest, we observe that every space ̂ M n is, of course, a complete direct summand of the space ̂ M∞ , hence, by Propositions 6.6 and 6.7(i), combined with Theorem 6.3, it is metrically projective. It remains to use Propositions 6.1 and 6.7(ii), and then Propositions 4.6(ii) and (again) 6.6.

162 | Helemskii Remark 6.9. Blecher [1] considered a different kind of projectivity, the operator space version of the classical ‘lifting property’ of Banach space (cf., e.,g., [13, p. 133]), studied by Grothendieck [6]. This notion also can be treated within the general frame-work of a rigged category and its free objects, but after a kind of elaboration of our scheme. Such an approach, in a frame-work of the so–called ‘asymptotic projectivity’, was used, in the context of operator spaces, in [8, 9]. For general matricially normed spaces, this approach leads to the following version of Theorem 3.10 in [1]: A matricially normed space is projective (in the just mentioned sense) if, and only if it is almost a direct summand of a ̂ M–composed space. As to the definition of an almost direct summand, its repeats word by word the Definition 3.8 in [1] that was given for operator spaces.

7 Some properties of the matricially normed ̂n space M In this section we again fix a natural n.

Theorem 7.1. The underlying normed space of ̂ M n is T n .

Proof. Denote the norm on M1 (̂ M n ) by || ⋅ ||∙ . Take an arbitrary element, say a, in ̂ M1 (M n ). First, we prove that ||a||∙ ≤ || ⋅ ||t . Accordingly, our task is to show that for every proper couple (E, v) we have ||φ v (a)|| ≤ ||a||t . Consider the commutative diagram (D2) with C as F and an arbitrary functional f on E as ψ. Fix v ∈ M n (E) and denote, for brevity, the matrix f n (v) ∈ M n (C) by b. Then for our a, as for a matrix in M n , we have that f[φ v (a)] = φ b (a). Consequently, knowing the map φ b : M n → C (cf. Section 3) and using the latter equation, we have f[φ v (a)] = tr(ab). Therefore the standard duality between M n and T n gives the estimate |f[φ v (a)]| ≤ ||b||o ||a||t . Now, using Hahn-Banach theorem, take f ∈ E∗ ; ||f|| = 1 such that f(φ v (a)) = v ||φ (a)||. Then, by the latter inequality, we have ||φ v (a)|| ≤ ||b||o ||a||t .

(7)

But it follows from [3, Cor. 3.3] that f , being considered as an operator between E and Cmin , is completely contractive. Since ||v|| ≤ 1, this implies that the norm of f n (v) in M n (Cmin ) = M n is also ≤ 1, that is ||b||o ≤ 1. Therefore the needed estimate for ||φ v (a)|| follows from (7). Turn to the inverse estimate. By the duality between M n and T n , there exists w ∈ M n ; ||w||o = 1 such that tr(aw) = ||a||t . Set E := Cmin and consider w in the

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unit ball in M n ⊗ Cmin . Then φ w : M n → C takes a to ||a||t . Therefore, by (2) (with m = 1), we have ||a||∙ ≥ ||a||t . Note that the underlying space of the operator space, playing in the smaller category of operator spaces the same role of ‘building bricks’ for free objects, is again T n [9, Prop. 2.7]. However, our current object, the space ̂ M n , is far away to be an operator space. We have already seen this for n = 1; now we demonstrate this for all n. Take p ∈ [1, ∞). A matricially normed space E is said to be p–convex or p– concave if, for every E–valued matrices u1 , . . . , u n (may be, of different sizes) we 1 1 have that ||u1 ⊕ ... ⊕ u n || ≤ ∑nk=1 ||u k ||p ) p or ||u1 ⊕ ... ⊕ u n || ≥ ∑nk=1 ||u k ||p ) p , respectively. A space that is p–convex and p–concave, is called an L p –space [3]. Evidently, an operator space is p–concave for every p. Proposition 7.2. The matricially normed space ̂ M n is not p–convex, in particular, not an L p –space, for every p > 1.

Proof. Suppose the contrary. Take any q–concave matricially normed space E ≠ 0 with 1 ≤ q < p (for example, Cmax ). Then, according to [15, Theorem 5.3] (cf. also [10, Prop. 3.3]), we have CC(̂ M n , E) = 0. On the other hand, B M n (E) has certainly more than one point. But, because of the freeness of ̂ M n , there is a bijection between the sets B M n (E) and CC(̂ M n , E). We came to a contradiction. Proposition 7.3. Let a ∈ M m (̂ M n ) be a blok–diagonal matrix a = (a1 ⊕...⊕a m ); a k ∈ ̂ M n . Then ||a||m ≥ 1n ∑k ||a k ||t .

Proof. As it is well known, there exist unitary matrices S k , T k : k = 1, . . . , m such that every S k a k T k is a positive diagonal matrix. Note that for S := (S1 ⊕ ... ⊕ S m ) and T := (T1 ⊕ ... ⊕ T m ) we have SaT = (S1 a1 T1 ⊕ ... ⊕ S m a m T m ). Therefore, by virtue of Axiom 2, we can suppose without loss of generality that all a k are positive diagonal matrices. Set E := Cmax , so we have M m (E) = T m ; m ∈ N. Also set v := 1n 1, where 1 is the identity matrix in ̂ M n , so we have v ∈ B M n (E) . It is easy to see that φ v : M n → C takes a to tr(va) = 1n tr(a), hence φ vm (a) is the diagonal m × m–matrix with numbers 1n tr(a k ) on the diagonal. Therefore, by (2), we have ||a|| ≥ ||φ vm (a)||t =

1 1 ∑ tr(a k ) = ||a k ||t . n k n

Note that Proposition 7.2 could be easily deduced from the previous proposition, without applying to the triviality of the space CC(̂ M n , E).

164 | Helemskii Proposition 7.3 shows, loosely speaking, that some properties of ̂ M n resemble (‘up to the multiplier 1n ’) those of L1 –spaces. Nevertheless we have Proposition 7.4. The matricially normed space ̂ M n ; n > 1 is not an L1 –space.

Proof. Suppose the contrary. As a particular case of Proposition 3.2 in [10], every functional f : E → Cmax , where E is an L1 –space, contractive in the ‘classical’ sense, is automatically completely contractive. Since the underlying space of ̂ Mn is T n , this concerns, in particular, f : ̂ M n → C : a 󳨃→ tr(a). Consequently, the operator f n : M n (̂ M n ) → M n (Cmax ) is contractive. In particular, for I ∈ M n (M n ) (see Section 5) we have ||f n (I)||t ≤ ||I||. But, by Proposition 5.4, we have ||I|| = 1, and at the same time f n (I) = ∑ij tr(e ij )e ji is the identity matrix in M n . Therefore ||f n (I)|| = n > 1, a contradiction.

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

D. P. Blecher, The standard dual of an operator space, Pacific J. Math., No. 1, 153 (1992), 15-30. D. P. Blecher, C. Le Merdy, Operator Algebras and their Modules, Clarendon Press, Oxford, 2004. E. G. Effros, Z.-J. Ruan, On matricially normed spaces, Pacific J. Math, 132, (1988), 24364. E. G. Effros, Z.-J. Ruan, Operator Spaces, Clarendon Press, Oxford, 2000. A. W. M . Graven, Injective and Projective Banach modules, Indagationes Mathematicae (Proceedings), 82 (1979), 253-272. A. Grothendieck, Une caracterisation vectorielle-metrique des espaces L1 , Canadian J. Math., 7 (1955), 552-561. C. Faith, Algebra: rings, modules and categories, vol. I, Springer-Verlag, BerlinHeidelberg-New York, 1973 A. Ya. Helemskii, Metric freeness and projectivity for classical and quantum normed modules, Sb. Math, 204 (2013), 1056-1083. A. Ya. Helemskii, Projectivity for operator modules: approach based on freedom, Rev. Roum. Math. Pures Appl. 59 (2014), A. Ya. Helemskii, Projective tensor product of protoquantum spaces, to appear in Colloc. Math., 2017. S. Mac Lane, Categories for the Working Mathematician. Springer-Verlag, Berlin, 1971. V. I. Paulsen. Completely Bounded Maps and Operator Algebras, Cam. Univ. Press, Cambridge, 2002. A. Pietsch, History of Banach Spaces and Linear Operators, Birkäuser, Boston-BaselBerlin, 2007. J. Pisier, Introduction to Operator Space Theory, Cam. Univ. Press, Cambridge, 2003. Z.-J. Ruan, Subspaces of C ∗ -algebras, J. Funct. Anal., 76 (1988), 217-230.

Bence Horvàth

Banach spaces whose algebras of operators are Dedekind-finite but they do not have stable rank one Abstract: In this note we examine the connection between the stable rank one and Dedekind-finite property of the algebra of operators on a Banach space X. We show that for the complex indecomposable but not hereditarily indecomposable Banach space X∞ constructed by Tarbard (Ph.D. Thesis, University of Oxford, 2013), the algebra of operators B(X∞ ) is Dedekind-finite but does not have stable rank one. While this sheds some light on the Banach space structure of X∞ itself, we observe that the indecomposable but not hereditarily indecomposable Banach space constructed by Gowers and Maurey (Math. Ann., 1997) does not possess this property. We also show that if K is the connected “Koszmider” space constructed by Plebanek in ZFC (Topology and its Applications, 2004), then B(C(K, R)) is Dedekind-finite but does not have stable rank one. Keywords: algebra of operators, Banach space, connected component, Dedekindfinite, indecomposable, Koszmider space, primary, stable rank one, Tarbard’s space. Classification: 47L10, 46H10, 46B07, 16D25.

1 Introduction and basic terminology Let A be a ring. We say that an element p ∈ A is idempotent if p2 = p holds. Let p, q ∈ A be idempotents, we say that p and q are equivalent, if there exist a, b ∈ A such that ab = p and ba = q. If p, q ∈ A are equivalent idempotents, we denote this by p ∼ q. It is easy to see that ∼ is an equivalence relation on the set of idempotents in A. Two idempotents p, q ∈ A are orthogonal if pq = 0 = qp. Definition 1.1. Let A be a unital ring with identity 1A . Then A is called (1) Dedekind-finite or directly finite or DF for short, if the only idempotent p ∈ A with p ∼ 1A is the identity 1A , (2) Dedekind-infinite if it is not Dedefind-finite,

Bence Horvàth , Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic. e-mail: [email protected] https://doi.org/10.1515/9783110602418-0009

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(3) properly infinite if there exist orthogonal idempotents p, q ∈ A such that p, q ∼ 1A . It is easy to see that a properly infinite ring is Dedekind-infinite. Clearly every commutative, unital ring is Dedekind-finite. Another easy example is the matrix ring M n (C), (n ≥ 1) since an (n × n) complex matrix is left-invertible if and only if it is right-invertible. Therefore it is natural to examine the unital Banach algebra B(X) from this perspective, where B(X) denotes the bounded linear operators on an infinitedimensional Banach space X. In this note all Banach spaces are assumed to be complex, unless explicitly stated otherwise. The systematic study of Dedekind(in)finiteness of B(X) was laid out by Laustsen in [11], where the author characterises Dedekind-finiteness and properly infiniteness of B(X) in terms of the complemented subspaces of X. For our purposes the former is of greater importance, therefore we recall this result here: Lemma 1.2. ([11, Corollary 1.5]) Let X be a Banach space. Then B(X) is Dedekindfinite if and only if no proper, complemented subspace of X is isomorphic to X as a Banach space. Let us recall that an infinite-dimensional Banach space X is indecomposable, if there are no closed, infinite-dimensional subspaces Y, Z of X such that X can be written as the topological direct sum X = Y ⊕ Z, this latter being a short-hand notation for X = Y + Z and Y ∩ Z = {0}. A Banach space X is hereditarily indecomposable (or HI for short) if every closed, infinite-dimensional subspace of X is indecomposable. As it is observed in [11, Corollary 1.7], every hereditarily indecomposable Banach space X satisfies the conditions of Lemma 1.2. However, as we shall demonstrate in Corollary 2.9, if X is an HI space, then B(X) in fact possesses the stronger property of having stable rank one. This definition was introduced by Rieffel in [16]: Definition 1.3. A unital Banach algebra A has stable rank one if the group of invertible elements inv(A) is dense in A with respect to the norm topology.

2 Algebras of operators with stable rank one and their connection to Dedekind-finiteness The following observation is an immediate corollary of [16, Proposition 3.1], we include a short proof for the reader’s convenience.

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Lemma 2.1. A unital Banach algebra with stable rank one is Dedekind-finite. Proof. Let A be a Banach algebra with stable rank one. Assume p ∈ A is idempotent such that p ∼ 1A , then there exist a, b ∈ A such that p = ab and 1A = ba. Let u ∈ inv(A) be such that ‖a − u‖ < ‖b‖−1 , then ‖1A − bu‖ = ‖ba − bu‖ ≤ ‖b‖‖a − u‖ < 1. So in particular bu ∈ inv(A) holds, and consequently b = buu−1 ∈ inv(A). From this and 1A = ba we get a = b−1 , consequently p = ab = 1A . Thus A is Dedekindfinite. Note however that the converse of the previous lemma is clearly false. We demonstrate this with an example which will be essential in the proof of our main result, Theorem 2.16. Let us recall that in a Banach algebra A an element a ∈ A is a topological zero divisor if inf{‖xa‖ + ‖ax‖ : x ∈ A, ‖x‖ = 1} = 0. It is a standard result from the fundamental theory of Banach Algebras, (see for example [2, Chapter 2, Theorem 14]) that for a unital Banach algebra A the topological boundary of inv(A), that is ∂(inv(A)) := inv(A)\inv(A), is contained in the set of topological zero divisors of A. In what follows N denotes the natural numbers excluding zero, and N0 := N ∪ {0}. Example 2.2. The unital Banach algebra ℓ1 (N0 ) (endowed with the convolution ∞

∞ product (a n )∞ n=0 ∗ (b n )n=0 := ∑ ∑ a k b m−k ) is Dedekind-finite but does not have m

m=0 k=0

stable rank one. The former is trivial since ℓ1 (N0 ) is commutative. Now let us show that it does not have stable rank one. This in fact is contained in the proof of [7, Proposition 4.7], we include the argument here for the sake of completeness. Let (δ n )n∈N0 stand for the canonical basis of ℓ1 (N0 ), clearly δ0 is the identity in ℓ1 (N0 ). Observe that δ1 is a non-invertible element in ℓ1 (N0 ). We now show that δ1 is not a topological zero divisor. To see this, let x = (x n )∞ n=0 ∈ ℓ1 (N0 ) be arbitrary. Then 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ‖x ∗ δ1 ‖ = 󵄩󵄩󵄩 ∑ x n−1 δ n 󵄩󵄩󵄩 = ∑ |x n | = ‖x‖. 󵄩󵄩n∈N 󵄩󵄩 n∈N 󵄩 󵄩 0

(1)

Thus by the discussion preceeding the example we see that δ1 ∉ ∂(inv(ℓ1 (N0 ))). Hence we conclude that δ1 ∉ inv(ℓ1 (N0 )), therefore ℓ1 (N0 ) cannot have stable rank one. As we will see in Corollary 2.9, all the examples given in [11] such that B(X) is Dedekind-finite have stable rank one. Thus the following question naturally arises:

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Question 2.3. Does there exist a Banach space X such that B(X) is Dedekind-finite but it does not have stable rank one? The purpose of the following is to answer this question in the positive. Recall that if A is a unital algebra over a field K and C is a unital subalgebra then inv(C) ⊆ inv(A) ∩ C holds but there is not equality in general. In the following, if J is a two-sided ideal of A we introduce the notation J ̃ := K1A + J. Lemma 2.4. Let A be an algebra over a field K and let J ⊴ A be a proper, two-sided ideal. Then for the unital subalgebra J ̃ the equality inv(J)̃ = inv(A) ∩ J ̃ holds.

Proof. It is clear that J ̃ is a unital subalgebra of A. Thus we only need to show the ̃ To see this let us pick an arbitrary λ ∈ K and j ∈ J such inclusion inv(A)∩ J ̃ ⊆ inv(J). that λ1A + j ∈ inv(A). Clearly λ ≠ 0 otherwise j ∈ inv(A) which contradicts J being a proper subset of A. Now it is clear that a := λ−1 1A − λ−1 (λ1A + j)−1 j ∈ K1A + J, and a simple calculation shows that a(λ1A + j) = 1A = (λ1A + j)a holds, proving the claim. Remark 2.5. If A is a complex unital Banach algebra and J ⊴ A is a proper, closed, two-sided ideal in A then J ̃ := C1A + J is a closed, unital subalgebra of A. (Closedness follows from the fact that C1A and J are respectively finite-dimensional and closed subspaces of the Banach space A.) Also, J ̃ is equal to the closed unital subalgebra in A generated by the set {1A } ∪ J. Lemma 2.6. Let A be a complex, unital Banach algebra and let a ∈ A be such that 0 ∈ C is not in the interior of the spectrum σ A (a). Then a ∈ inv(A).

Proof. By the hypothesis it follows that 0 ∉ int(σ A (a)) = C\ (C\σ A (a)). Thus there exists a sequence (λ n )n∈N in the resolvent set of the element a converging to 0 ∈ C. Therefore (a − λ n 1A )n∈N is a sequence of invertible elements in A converging to a. An operator T ∈ B(X) is called inessential if for any S ∈ B(X) it follows that dim(Ker(I X − ST)) < ∞ and codimX (Ran(I X − ST)) < ∞. The set E(X) of inessential operators forms a proper, closed, two-sided ideal of B(X), see [14, Remark 4.3.5]. Proposition 2.7. Let X be a Banach space, and let J ⊴ B(X) be a closed, two-sided ideal with J ⊆ E(X). Then for any α ∈ C and T ∈ J, αI X + T ∈ inv(J)̃ holds, and therefore J ̃ has stable rank one. Proof. Let us pick α ∈ C and T ∈ J arbitrary. It is an immediate corollary of Lemma 2.4 that σ J ̃ (T) = σB(X) (T). Now by the Spectral Mapping Theorem σ J ̃ (αI X + T) =

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α + σ J ̃ (T), putting this together with the previous we conclude that σ J ̃ (αI X + T) = α + σB(X) (T).

(2)

Since T ∈ J ⊆ E(X), it follows from [3, Lemma 5.6.1] that T is a Riesz operator (see [3, Definition 3.1.1]), thus σB(X) (T)\{0} has no accumulation point, thus σB(X) (T) must be countable. Consequently σ J ̃ (αI X +T) must be countable, thus it has empty ̃ interior, so in particular Lemma 2.6 yields αI X + T ∈ inv(J). Remark 2.8. Let us note that in the previous proposition the assumption that the ideal is contained in the inessential operators cannot be dropped in general. To see this we consider the pth quasi-reflexive James space Jp , where 1 < p < ∞. Since the closed, two-sided ideal W(Jp ) of weakly compact operators is onecodimensional in B(Jp ), it is in particular a complemented subspace of B(Jp ) and therefore B(Jp ) = C I Jp + W(Jp ) holds. On the other hand, as observed in [11, Propostition 1.13], the Banach algebra B(Jp ) is Dedekind-infinite so by Lemma 2.1 it cannot have stable rank one. On a Banach space X an operator T ∈ B(X) is called strictly singular if there is no infinite-dimensional subspace Y of X such that T|Y is an isomorphism onto its range. The set of strictly singular operators on X is denoted by S(X) and it is a closed, two-sided ideal in B(X). By [3, Theorem 5.6.2] the containment S(X) ⊆ E(X) also holds. Corollary 2.9. For a complex hereditarily indecomposable Banach space X the Banach algebra B(X) has stable rank one. Proof. As it was proven by Gowers and Maurey in [8, Theorem 18], for any complex HI space B(X) = C I X + S(X) holds. Together with Proposition 2.7 the result immediately follows. The result above is known, see for example in [7] and the text preceeding Theorem 4.16, although deduced in a slightly different way to ours. The following simple algebraic lemma is the key step in the proof our main result. Lemma 2.10. Let A be a unital algebra over a field K and let J ⊴ A be a two-sided ideal such that both J ̃ and A/J are Dedekind-finite. Let π : A → A/J denote the quotient map. If π [inv(A)] = inv (A/J) holds then A is Dedekind-finite.

Proof. Let p ∈ A be an idempotent such that p ∼ 1A . Then there exist a, b ∈ A such that ab = 1A and ba = p. The identities π(a)π(b) = π(1A ) and π(b)π(a) = π(p) show that π(p) is an idempotent in A/J such that π(p) ∼ π(1A ). Since A/J is DF it follows that π(p) = π(1A ), equivalently π(b)π(a) = π(1A ) and consequently

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π(a) ∈ inv (A/J). By the assumption π restricted to the set of invertible elements of A surjects onto the set of invertible elements of A/J therefore there exists c ∈ inv(A) such that π(a) = π(c), equivalently a − c ∈ J. Thus c−1 − b = c−1 ab − c−1 cb = c−1 (a − c)b ∈ J.

Let us define a󸀠 := (a − c)c−1 and b󸀠 := c(b − c−1 ), it is clear from the previous that a󸀠 , b󸀠 ∈ J. Now we show that the following identities hold: – (1A + a󸀠 )(1A + b󸀠 ) = 1A or equivalently a󸀠 + b󸀠 + a󸀠 b󸀠 = 0, – (1A + b󸀠 )(1A + a󸀠 ) = cpc−1 or equivalently a󸀠 + b󸀠 + b󸀠 a󸀠 = cpc−1 − 1A . To see these, we observe that from the definitions of a󸀠 and b󸀠 we obtain a󸀠 + b󸀠 = (a − c)c−1 + c(b − c−1 ) = ac−1 + cb − 2 ⋅ 1A ,

(3)

b󸀠 a󸀠 = c(b − c−1 )(a − c)c−1 = c(ba − bc − c−1 a + 1A )c−1 = cpc−1 − cb − ac−1 + 1A , (4) a󸀠 b󸀠 = (a − c)c−1 c(b − c−1 ) = ab − ac−1 − cb + 1A = 2 ⋅ 1A − ac−1 − cb.

(5)

The above immediately yield the required identities. Thus we obtained that cpc−1 is an idempotent in J ̃ equivalent to 1A . Since J ̃ is DF it follows that cpc−1 = 1A . This is, p = 1A which concludes the proof. In what follows, if K is a compact Hausdorff space then C(K) denotes the complex valued continuous functions on K. If X is a Banach space then K(X) denotes the closed, two-sided ideal of compact operators. By [3, Theorems 4.4.4 and 5.6.2] the containment K(X) ⊆ E(X) holds. Remark 2.11. Let us note here that in the previous lemma, the condition that the invertible elements in A surject onto the invertible elements in A/J is not superfluous. To see this, we recall some basic properties of the Toeplitz algebra, see [18, Example 9.4.4] for full details of the construction. Let H be a separable Hilbert space and let S ∈ B(H) be the right shift operator, let S∗ ∈ B(H) denote its adjoint. The unital sub-C∗ -algebra of B(H) generated by S is called the Toeplitz-algebra T. We recall that K(H) ⊆ T and that T/K(H) is isomorphic to C(T), where T is the unit circle. Since C(T) is commutative, it is clearly Dedekind-finite. As is well̃ known, (see [4, Corollary 5] or by Proposition 2.7 above) K(H) has stable rank one thus by Lemma 2.1 it is also Dedekind-finite. On the other hand, S∗ S = IH and SS∗ ≠ IH , thus T is Dedekind-infinite.

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For a unital Banach algebra A let exp(A) := {exp(a) : a ∈ A}. Recall that exp(A) ⊆ inv(A) and when A is commutative, exp(A) is both a subgroup and the connected component of the identity in inv(A). In other words, exp(A) is the maximal connected subset of inv(A) -ordered by inclusion- containing 1A . For further details we refer the reader to [5, Corollary 2.4.27]. Lemma 2.12. Let A be a unital Banach algebra and suppose J ⊴ A is a closed, twosided ideal in A such that A/J is commutative. Let π : A → A/J denote the quotient map. If inv(A/J) is connected then π [exp(A)] = inv(A/J) holds. In particular π [inv(A)] = inv(A/J).

Proof. Since A/J is commutative and inv(A/J) is connected it follows that inv(A/J) = exp(A/J). We now observe that exp(A/J) = π[exp(A)] holds, since for any a ∈ A, the series expansion of exp(a) converges (absolutely) in A and the quotient map π is a continuous algebra homomorphism; thus it readily follows that π(exp(a)) = exp(π(a)). The second part of the claim follows from π[inv(A)] ⊆ inv(A/J). Lemma 2.13. The group inv(ℓ1 (N0 )) is connected.

Proof. Let A := ℓ1 (N0 ). It is known (see for example [5, Theorem 4.6.9]) that the character space Γ A of A is homeomorphic to the closed unit disc D. Thus by the Arens–Royden theorem (see [13, 3.5.19 Theorem] and the text preceeding it) we obtain the following isomorphism of groups: inv(A)/exp(A) ≃ inv(C(D))/exp(C(D)) ≃ π 1 (D),

(6)

where π 1 (D) denotes the first fundamental group of D. Since D is simply connected we obtain inv(A) = exp(A) proving that inv(A) is connected as required. Remark 2.14. In the proof of the previous lemma we do not use the surjective part of the Arens–Royden theorem, only the much weaker statement that inv(A)/exp(A) injects into inv(C(Γ A ))/exp(C(Γ A )). Let us recall the properties of Tarbard’s ingenious indecomposable Banach space construction that are relevant to our purposes, we refer the interested reader to [19, Chapter 4] to see the following theorem in its full might. Theorem 2.15. ([19, Theorem 4.1.1]) There exists an indecomposable Banach space X∞ such that the unital Banach algebras B(X∞ )/K(X∞ ) and ℓ1 (N0 ) are isometrically isomorphic. We are now ready to state and prove the main result of this note. Theorem 2.16. The Banach algebra B(X∞ ) is Dedekind-finite but does not have stable rank one.

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Proof. We first show that B(X∞ ) does not have stable rank one. Assume towards a contradiction that it does. Then it immediately follows that B(X∞ )/K(X∞ ) also has stable rank one, which in view of Theorem 2.15 is equivalent to ℓ1 (N0 ) having stable rank one. This is impossible by Example 2.2. Now we show that B(X∞ ) is Dedekind-finite. By Proposition 2.7 we obtain that ̃ ∞ ) has stable rank one so by Lemma 2.1 it is Dedekind-finite. By Example 2.2 K(X we have that ℓ1 (N0 ) and thus B(X∞ )/K(X∞ ) is also Dedekind-finite. Thus applying Lemmas 2.13, 2.12 and 2.10 successively, we obtain that B(X∞ ) is Dedekindfinite, which completes the proof. With the aid of Lemma 1.2 we observe the following: Corollary 2.17. No proper, complemented subspace of X∞ is isomorphic to X∞ .

We do not know if there is an entirely Banach space-theoretic proof of this result. However, we would like to draw the reader’s attention to the fact that the previous corollary does not hold in general for indecomposable Banach spaces. This follows directly from a deep result of Gowers and Maurey [9]. We recall that an infinite-dimensional Banach space X is prime if it is isomorphic to all its infinite-dimensional, complemented subspaces. Theorem 2.18. ([9, Section (4.2) and Theorem 13]) There exists an indecomposable, prime Banach space. In fact, with the help of two easy lemmas we can say a bit more. In order to do this let us recall the following well-known result, see for example the second part of [11, Corollary 1.5]: Lemma 2.19. Let X be a Banach space, let P, Q ∈ B(X) be idempotents. Then P ∼ Q if and only if Ran(P) ≃ Ran(Q). Lemma 2.20. Let X be an indecomposable Banach space. Then B(X) cannot be properly infinite. Proof. Assume towards a contradiction that B(X) is properly infinite. Then there exist P, Q ∈ B(X) orthogonal idempotents such that P, Q ∼ I X . By Lemma 2.19 this is equivalent to Ran(P) ≃ X ≃ Ran(Q). Clearly X = Ran(P) ⊕ Ran(I X − P) and since Ran(P) is infinite-dimensional and X is indecomposable we obtain that Ran(I X − P) must be finite-dimensional. Consequently, the range of Q = Q(I X − P) is finite-dimensional, contradicting Ran(Q) ≃ X.

An infinite-dimensional Banach space X is primary if for every P ∈ B(X) idempotent either Ker(P) or Ran(P) is isomorphic to X. A prime Banach space is clearly primary.

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Lemma 2.21. Let X be a primary Banach space. Then B(X) is Dedekind-infinite. Proof. Let P ∈ B(X) be an idempotent with dim(Ker(P)) = 1. Since X is primary, Ran(P) ≃ X holds. By Lemma 2.19 this is equivalent to P ∼ I X . If B(X) were DF then P = I X which is impossible. Theorem 2.18 ensures that the following corollary of Lemmas 2.20, 2.21 is not vacuous: Corollary 2.22. For an indecomposable, primary Banach space X the algebra of operators B(X) is Dedekind-infinite but not properly infinite. Remark 2.23. Let K be a countable compact metric space. By a deep result [12, Theorem B] of Motakis, Puglisi and Zisimopoulou, there exists a Banach space X K such that B(X K )/K(X K ) ≃ C(K). In [16], Rieffel introduced the notion of left stable rank in general for unital Banach algebras. We observe here that B(X K ) has left stable rank 1 or 2, in the sense of Rieffel. By [16, Corollary 4.12], to show this, it ̃ K ) and C(K) have both stable rank one. The former is enough to observe that K(X follows from Proposition 2.7. For the latter, in view of [16, Proposition 1.7], it is enough to see that K is zero-dimensional. This in particular follows if K is totally disconnected, by [1, Proposition 3.1.7]. But this is clearly holds, since K is countable and compact. We do not know however if for certain K-s the Banach algebra B(X K ) is DF but does not have stable rank one; our “lifting invertible elements” method is not applicable here, since K is totally disconnected. After reading the first draft of this paper, it was suggested to us by Piotr Koszmider that the C(K)-space from [15] is an example of a real Banach space such that its algebra of operators is Dedekind-finite but it does not have stable rank one. With his kind permission we include a proof here. In the following, if K is a compact Hausdorff space, we always consider C(K) as a real Banach space. To emphasise this, we write C(K, R) for C(K). Let K be a compact Hausdorff space, let g ∈ C(K, R) then M g : C(K, R) → C(K, R);

f 󳨃→ fg

(7)

is called the multiplication operator. An operator T ∈ B(C(K, R)) is called a weak multiplication if there is a g ∈ C(K, R) and S ∈ S(C(K, R)) such that T = M g + S. We say that an infinite compact Hausdorff space is a Koszmider space if every bounded linear operator on C(K, R) is a weak multiplication. In [10, Theorem 6.1] Koszmider showed that assuming the Continuum Hypothesis, both connected and zero-dimensional Koszmider spaces exist. In [15, Theorem 1.3] Plebanek showed the existence of a connected Koszmider space without any assumptions beyond ZFC. We recall the following two results on Koszmider spaces:

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Theorem 2.24. ([6, Theorem 6.5(i)]) Let K be a Koszmider space without isolated points. Then B(C(K, R))/S(C(K, R)) and C(K, R) are isomorphic as unital Banach algebras. The following result is an immediate corollary of Theorems 2.3 and 5.2 in Koszmider’s paper [10]. Theorem 2.25. ([10]) Let K be a connected Koszmider space. Then C(K, R) is not isomorphic to any of its proper, closed subspaces. Lemma 2.26. Let K be a compact connected Hausdorff space with at least two points. Then C(K, R) does not have stable rank one. Proof. Let x, y ∈ K be distinct. Since K is Hausdorff, we can take U, V disjoint open subsets of K such that x ∈ U and y ∈ V. By Urysohn’s lemma there exist f, g ∈ C(K, R) supported on U and V, respectively, such that f(x) = 1 and g(y) = 1. Let h := f − g, clearly h ∈ C(K, R) is such that h(x) = 1 and h(y) = −1. Suppose k ∈ C(K, R) is such that ‖k − h‖ < 1/2, thus |k(x) − 1| < 1/2 and |k(y) + 1| < 1/2. In particular, 1/2 < k(x) and −1/2 > k(y), thus by continuity of k and connectedness of K we obtain that there is z ∈ K such that k(z) = 0. Thus k ∈ C(K, R) cannot be invertible. This shows that inv(C(K, R)) is not dense in C(K, R), as required. Remark 2.27. We note however that Lemma 2.26 is not true in general for complex C(K)-spaces; indeed, [0, 1] is one-dimensional, so by [16, Proposition 1.7] C([0, 1], C) has stable rank one. Proposition 2.28. Let K be a connected Koszmider space. Then B(C(K, R)) is Dedekind-finite but it does not have stable rank one. Proof. Assume towards a contradiction that B(C(K, R)) has stable rank one. Then B(C(K, R))/S(C(K, R)) also has stable rank one, which in view of Theorem 2.24 is equivalent to C(K, R) having stable rank one. This is impossible by Lemma 2.26. The fact that B(C(K, R)) is DF follows from Lemma 1.2 and Theorem 2.25. Remark 2.29. We remark in passing that both the real and complex examples share the following property: Tarbard’s space X∞ and C(K, R) (where K is a connected Koszmider space) are both indecomposable, but not hereditarily indecomposable Banach spaces. Indeed, X∞ is indecomposable by [19, Proposition 4.1.5] but not hereditarily indecomposable by [19, Proposition 4.1.4]. If K is a connected Koszmider space then by [10, Theorem 2.5] it follows that C(K, R) is indecomposable. On the other hand, it is well-known that for any infinite compact Hausdorff space K, C(K) cannot be hereditarily indecomposable. This follows from the fact that if K is such then C(K) has a closed subspace (isometrically) isomorphic to c0 , see for example, [17, Lemma 2.5(d)].

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Acknowledgement: The author is grateful to his supervisors Dr Yemon Choi and Dr Niels J. Laustsen (Lancaster) for their invaluable advice during the preparation of this note. He is indebted to Dr Tomasz Kania (Prague), Professor Piotr Koszmider (Warsaw) and the anonymous referee for the encouragement and for carefully reading the first version of this note. He also acknowledges the financial support from the Lancaster University Faculty of Science and Technology and acknowledges with thanks the partial funding received from GAČR project 19-07129Y; RVO 67985840 (Czech Republic).

Bibliography [1] [2] [3] [4] [5] [6]

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

A. Archangel‘skii and M. Tkachenko. Topological Groups and Related Structures. Atlantis Press, Paris, 2008. F. F. Bonsall and J. Duncan. Complete Normed Algebras. Springer-Verlag, New York, 1973. S. R. Caradus, W. E. Pfaffenberger, and B. Yood. Calkin Algebras and Algebras of Operators on Banach Spaces. Marcel Dekker, Inc., New York, 1974. G. Corach and A. R. Larotonda. Stable range in Banach algebras. J. Pure. Appl. Alg., 32:289–300, 1984. H. G. Dales. Banach Algebras and Automatic Continuity. Oxford University Press Inc., New York, 2000. H. G. Dales, T. Kania, T. Kochanek, P. Koszmider, and N. J. Laustsen. Maximal left ideals of the Banach algebra of bounded linear operators on a Banach space. Studia Math., 218(3):245–286, 2013. Sz. Draga and T. Kania. When is multiplication in a Banach algebra open? Lin. Alg. Appl., 538:149–165, 2018. W. T. Gowers and B. Maurey. The unconditional basic sequence problem. J. Amer. Math. Soc., (6):851–874, 1993. W. T. Gowers and B. Maurey. Banach spaces with small spaces of operators. Math. Annalen, 307(4):543–568, 1997. P. Koszmider. Banach spaces of continuous functions with few operators. Math. Ann., 330(1):151–183, 2004. N. J. Laustsen. On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces. Glasgow Math. J., 45(1):11–19, 2003. P. Motakis, D. Puglisi, and D. Zisimopoulou. A hierarchy of Banach spaces with C(K) Calkin algebras. Indiana Univ. Math. J., 65(1):39–67, 2016. Th. W. Palmer. Banach Algebras and the General Theory of ∗-Algebras. Cambridge University Press, Cambridge, 1994. A. Pietsch. Operator Ideals. North-Holland Publishing Company, 1979. G. Plebanek. A construction of a Banach space C(K) with few operators. Topology and its Applications, 143(1):217–239, 2004. M. A. Rieffel. Dimension and stable rank in the K-theory of C ∗ -algebras. Proc. London Math. Soc., s346(2):301–333, 1983.

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[17] H. P. Rosenthal. The Banach Spaces C(K) (Handbook of the Geometry of Banach Spaces, Vol.2). Elsevier Science B.V., 2003. [18] M. Rørdam, F. Larsen, and N. J. Laustsen. An Introduction to K-Theory for C ∗ -algebras. Cambridge University Press, New York, 2000. [19] M. Tarbard. Operators on Banach Spaces of Bourgain-Delbaen Type. Ph.D. Thesis, University of Oxford, 2013.

Anthony To-Ming Lau

Invariant Complementation Property and Fixed Point Sets On Power Bounded Elements in the Group von Neumann Algebra Dedicated to Professor Eberhard Kaniuth for his warm friendship and research collaboration Abstract: In this paper, we discuss the invariant complementation property of the group Von Neuman algebra VN(G) generated by the left translation operators on L2 (G) of a locally compact group G, and the fixed point set of power bounded elements in VNG). Keywords: amenable group, group von Neumann algebra, complementation property, power bounded elements, Fourier and Fourier Stieltjes algebras, positive definite functions. Classification: 46H20, 43A20 and 43A10.

1 Introduction A classical result of H. Rosenthal [20] shows that if G is a locally compact abelian group, X is weak∗ -closed translation invariant subspace of a L∞ (G), and X is complemented in L∞ (G), then X is invariantly complemented, i.e. X admits a translation invariant closed complement, or equivalently X is the range of a continuous projection on L∞ (G) commuting with translations. Later in [15], the author shows that a locally compact group G is amenable if and only if every weak∗ -closed left translation invariant subalgebra M which is closed under conjugation in L∞ (G) is invariantly complemented. It is the purpose of this paper to discuss the invariant complementation property of the group von Neumann algebra VN(G) of a locally compact group G, and the fixed point set of power bounded elements in VN(G).

This research is supported by NSERC Grant ZC912. Anthony To-Ming Lau, Department of Mathematical and Statistical Sciences, University of Alberta Edmonton, Alberta, Canada T6G 2G1, e-mail address: [email protected] https://doi.org/10.1515/9783110602418-0010

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2 Some preliminaries Let G be a locally compact group with a fixed left Haar measure λ. Let L1 (G) be the group algebra of G with convolution product. We define C∗ (G), the group C∗ -algebra of G, to be the completion of L1 (G) with respect to the norm ‖f‖∗ = sup ‖π f ‖, where the supremum is taken over all nondegenerate ∗-representations π of L1 (G) as a ∗-algebra of bounded operators on a Hilbert space. Let B(L2 (G)) be the set of all bounded operators on the Hilbert space L2 (G) and ρ be the left regular representation of G, i.e., for each f ∈ L1 (G), ρ(f) is the bounded operator in B(L2 (G)) defined by ρ(f)(h) = f ∗ h, the convolution of f and h in L2 (G). Denote by C∗ρ (G) the completion of L1 (G) with respect to the norm ‖ρ(f)‖, f ∈ L1 (G), and denote by VN(G) the closure of {ρ(f) : f ∈ L1 (G)} in the weak operator topology in B(L2 (G)). In the case when G is amenable, which is the case when G is compact, then C∗ (G) is isometric isomorphic to C∗ρ (G). Denote the set of continuous positive definite functions on G by P(G), and the set of continuous functions on G with compact support by C00 (G). We define the Fourier-Stieltjes algebra of G, denoted by B(G), to be the linear span of P(G). Then B(G) is a Banach algebra with the norm of each ϕ ∈ B(G) defined by ‖ϕ‖ =

󵄨 󵄨󵄨 󵄨󵄨 ∫ f(t)ϕ(t)dλ(t)󵄨󵄨󵄨. 󵄨 󵄨󵄨 f ∈L1 (G),‖f‖∗ ≤1 󵄨 sup

The Fourier algebra of G, denoted by A(G), is defined to be the closed linear span of P(G) ∩ C00 (G). Clearly, A(G) = B(G) when G is compact. It is known that C∗ (G)∗ = B(G), where the duality is given by ⟨f, ϕ⟩ = ∫ f(t)ϕ(t)dλ(t), f ∈ L1 (G), ϕ ∈ B(G), and A(G)∗ = VN(G). A locally compact group G is called amenable if there is a left invariant mean m on CB(G), the space of bounded continuous complex-valued functions on G, i.e. m ∈ CB(G)∗ , ‖m‖ = m(1) = 1 and ⟨m, ℓa f⟩ = ⟨m, f⟩ for all f ∈ CB(G), a ∈ G, where (ℓa f)(x) = f(ax), x ∈ G. In this case, ‖f‖∗ = ‖ρ(f)‖ for all f ∈ L1 (G). See also the recent book [17] of the author and E. Kaniuth on the Fourier and Fourier Stieltjes algebra of locally compact groups for more details.

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3 Invariant complementation property of the group von Neumann algebra Let G be a locally compact group. Then there is a natural action of A(G) on VN(G) defined by ⟨ϕ ⋅ x, ψ⟩ = ⟨x, ϕψ⟩, x ∈ VN(G). When G is commutative, then A(G) and VN(G) are isometrically isomorphic to L1 (̂ G) and L∞ (̂ G) respectively (where ̂ G is the dual group of G) and the action of A(G) on VN(G) corresponds to convolution of functions in L1 (̂ G) and L∞ (̂ G), (See [14] for more details.) A subspace M of VN(G) is called invariant if ϕ ⋅ x ∈ M for all ϕ ∈ A(G), x ∈ M. Define ∑ (M) = {g ∈ G; l g ∈ M}. If M is an invariant W ∗ -subalgebra of VN(G), then Σ(M) = H is a non-empty closed subgroup of G and M = N H , the ultraweak closure of the linear span of {l g ; g ∈ H} in VN(G) (see [18, Theorems 6 and 8]). The following was proved in [16]:

Theorem 3.1. Let M be an invariant W ∗ -subalgebra of VN(G) such that Σ(M) = H is a normal subgroup of G. Then there exists a continuous projection P of VN(G) onto M such that P(ϕ ⋅ x) = ϕ ⋅ P(x) for all ϕ ∈ A(G) and x ∈ VN(G). In particular, M admits a closed complement which is also invariant. Let H be closed subgroup of G and VN H (G) = ⟨ρ(h) : h ∈ H⟩

WOT

⊆ VN(G)

where WOT denotes the weak operator topology on B(L2 (G)). Then VN H (G) is an invariant W ∗ -subalgebra of VN(G). The following result was proved by TakesatiTatsuma in [23]: Theorem 3.2. If M is an invariant W ∗ -subalgebra of VN(G), then weak∗

M = ⟨ρ(x) : x ∈ H⟩

= VN H (G),

where H = Σ(M) and ρ(x)h(y) = h(x−1 y), h ∈ L2 (G), y ∈ G. Hence there is a 1 − 1 correspondence between closed subgroups H of G and invariant W ∗ -algebra of VN(G). Let [SIN] denote the class of all locally compact groups such that there is a basis of neighbourhood basis of the identity consisting of compact sets V such that x−1 Vx = V for all x ∈ G. The class of all [SIN]-groups include all compact groups, discrete groups and abelian groups.

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A locally compact group G is said to have the complementation property if every weak∗ -closed invariant W ∗ -subalgebra of VN(G) is invariantly complemented. The following theorem follows from a result of Forrest [3] and Proposition 3.1 in [7]. Theorem 3.3. Every [SIN]-group has the complementation property. However, the Heisenberg group has the complementation property but it is not a [SIN]-group. For a closed subgroup H of a locally compact group G, let P1 (G) = {ϕ ∈ P(G); ϕ(e) = 1}

P H (G) = {ϕ ∈ P(G); ϕ(h) = 1 for all

h ∈ H}

Then P H (G) ⊆ P1 (G) and P H (G) with pointwise multiplication is a commutative semigroup. We call H a separating subgroup if for any x ∈ G\H, there exists ϕ ∈ P H (G) such that ϕ(x) ≠ 1. A locally compact group G is said to have the separation property if each closed subgroup of G is separating. Examples of separating subgroups include all open subgroups, compact groups and normal subgroups. Theorem 3.4 ([3]). Every SIN-group has the separation property. Example 3.5. G = affine group of the real line = 2 × 2 matrices of form {(

a 0

s ) : a > 0, s ∈ IR} ←→ {(a, s); a > 0, s ∈ IR} 1 (a, s)(b, t) = (ab, s + at).

Let H = {(a, 0); a > 0}. Then H is not separating. Indeed, if ϕ ∈ P H (G), x, y ∈ G Hence

|ϕ(xy) − ϕ(x)ϕ(y)|2 ≤ (1 − |ϕ(x)|2 )(1 − |ϕ(y)|2 ). ϕ(h1 xh2 ) = ϕ(x)

For t > 0, x t = (1, t) we have

∀ x ∈ G,

h1 , h2 ∈ H.

H(1, t)H = G+ = {(a, s); a > 0, s > 0}.

Hence: so by continuity, t → 0+

ϕ(h1 x t h2 ) = ϕ(x t ) ϕ(g) = 1 for all

∀ h1 , h2 ∈ H g ∈ G+ .

Similarly, by considering t < 0, we have ϕ(g) = 1 ϕ = 1.

for all

g ∈ G− . Consequently

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Example 3.6. Let G be the Heisenberg group, i.e. G = all 1 [ [0 [0

x 1 0

3 × 3 matrices

z ] y ] ←→ (x, y, z) 1]

(x1 , y1 , z1 )(x2 , y2 , z2 ) = (x1 + x2 , y1 + y2 , z1 + z2 + x1 y2 ) Centre of G = Z(G) = {(0, 0, t); t ∈ IR}.

Let H = {(x, 0, 0); x ∈ IR} < G. Then H is not separating. Indeed, let ϕ ∈ P H (G). For y ≠ 0, let g y = (0, y, 0). Then

{hg y h−1 g −1 y ; h ∈ H} = {(0, 0, t) : t ∈ IR} = Z(G). Since

−1 −1 ϕ(g y ) = ϕ(hg y h−1 ) = ϕ( (hg y h g y ) ⋅g y ) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ∈Z(G)

we obtain that ϕ(g y ) = ϕ(g y ⋅ g) ∀ g ∈ Z(G)

y ≠ 0, y ∈ IR.

With y → 0, we conclude that ϕ(g) = 1 ∀ g ∈ Z(G). The following theorem was proved in [7]. Theorem 3.7. (a) For any locally compact group G, separation property implies invariant complementation property. (b) Let G be a connected locally compact group. Then G has the separation property if and only if G is a [SIN]-group. A locally compact group is called an [IN]-group if there is a neighbourhood V such that x−1 Vx = V for all x ∈ G. Remark 3.8. Losert showed that there is an example of a locally compact group G such that G has a compact open normal subgroup and every proper closed subgroup of G is compact (in particular, G is an [IN]-group) with the separation property and hence the invariant complementation property but G is not a SIN-group.

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4 Fixed point sets of power bounded elements in VN(G) Let G be a locally compact group. For σ ∈ B(G), T ∈ VN(G), define σ ⋅ T ∈ VN(G) ⟨σ ⋅ T, ψ⟩ = ⟨T, σψ⟩, ψ ∈ A(G). Let

I σ = {σϕ − ϕ; ϕ ∈ A(G)}−‖⋅‖ ⊆ A(G).

Then (i) I σ is a closed ideal in A(G) (ii) I σ⊥ = {T ∈ VN(G) : σ ⋅ T = T}. Elements in I σ⊥ are called σ-harmonic functions on A(G) (see [1], [2]) and it is a weak∗ -closed invariant subspace of VN(G). For a discrete group D, let R(D) denote the Boolean ring of subsets of D generated by all left cosets of subgroups of D. Let R c (G) = {E ∈ R(G d ) : E is closed in G} G d = denote G with the discrete topology. For any locally compact group G and u ∈ B(G), define: E u = {x ∈ G : |u(x)| = 1}

and

F u = {x ∈ G : u(x) = 1}.

The following result is due to J. Gilbert [5], B. Schrieber [22] for the abelian case and B. Forrest [3] for the general case. Theorem 4.1. Let G be a locally compact group. Then E ∈ R c (G) if and only if E = ∪ (a i H i \ ∪ b i,j K ij ), where a i , b i,j ∈ G, H i is a closed subgroup of G and K ij is an n

mi

i=1

j=1

open subgroup of H i .

Let G and H be groups. A map α : C ⊆ G → H is called affine if C is a coset and for any r, s, t ∈ C, α(rs−1 t) = α(r)α(s)−1 α(t).

A map α : Y ⊆ G → H is called piecewise affine if (i) there exist pairwise disjoint sets Y i ∈ R(G), i = 1, . . . , n, such that Y = n

∪ Yi ,

i=1

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(ii) each Y i is contained in a coset C i on which there is an affine map α i : C i → H such that α i |Y i = α|Y i . The following theorem is due to Illie and Spronk [6]: Theorem 4.2. Let G and H be locally compact groups with G amenable, and let Φ : A(G) → B(H) be a completely bounded homomorphism. Then there is a continuous piecewise affine map α : Y ⊂ H → G such that for each h in H {u(α(h)) Φ u (h) = { 0 {

if

h ∈ Y,

otherwise.

Lemma 4.3. Let G be a locally compact group and u a power bounded element of B(G) such that E u is open in G. Then u|E u is a piecewise affine map from E u into T . The following theorem is proved in [11]. Theorem 4.4. Let G be any locally compact group and u ∈ B(G) be power bounded (i.e. sup{‖u n ‖; n = 1, 2, . . . } < ∞). Then (a) The sets E u and F u are in R c (G). (b) The fixed point set of u in VN(G) = {T ∈ VN(G); u ⋅ T = T} is the range of a projection P : VN(G) → VN(G) such that u ⋅ P(T) = P(u ⋅ T) for all T ∈ VN(G). W∗ If G is amenable, then {T ∈ VN(G); u ⋅ T = T} = ⟨ρ(x); x ∈ F u ⟩ . Note: When G is abelian, (a) is due to B. Schrieber. The next theorem is due to Bert Schreiber for G abelian [22]. It is proved in [12] in the general case. Theorem 4.5. Let G be a locally compact group and let u be a power bounded element of B(G). Then there exist closed subsets F1 , . . . , F n of G with the following properties: n

(1) F j ∈ Rc (G), 1 ≤ j ≤ n, and E u = ∪ F j . j=1

(2) For each j = 1, . . . , n, there exist a closed subgroup H j of G, a j ∈ G, α j ∈ T and a continuous character γ j of H j such that F j ⊆ a j H j and u(x) = α j γ j (a−1 j x) for all

x ∈ Fj .

Proof. Consider the group G equipped with the discrete topology. Let i : G d → G denote the identity map. Then u ∘ i ∈ B(G d ) and ‖u ∘ i‖B(G d ) = ‖u‖B(G) and hence u ∘ i is power bounded. Therefore, by Lemma 4.3. there exist subsets S i of G, subgroups L i of G, c i ∈ G and affine maps β i : c i L i → T , i = 1, . . . , r, with the following properties:

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n

(1) S i ∈ R(G d ) and E u = ∪ S i ; i=1

(2) For each i = 1, . . . , n, S i ⊆ c i L i and β i |S i = u|S i . Now each S i is of the form ⋃ dℓ (Mℓ \ ⋃ eℓk Nℓk ), q

qℓ

ℓ=1

k=1

where dℓ , eℓk ∈ G, the Mℓ are subgroups of G and the Nℓk are subgroups of Mℓ , 1 ≤ ℓ ≤ q, 1 ≤ k ≤ qℓ . Thus, by a further reduction step, we can assume that we have S = c(A\ ⋃ B j ) m

j=1

where B j is either empty or a coset in A. In addition, since K j has infinite index in H and A has finite index in H, the subgroup corresponding to B j has infinite index in A. Since u ∈ B(G) is uniformly continuous, the affine map δ : cA → T is uniformly continuous as well and hence extends to a continuous affine map δ : cA → T . Then δ agrees with u on S since u is continuous. Let γ denote the continuous character of A associated with δ. Then u(x) = αγ(c−1 x) for all x ∈ S. Finally, since E u is closed in G, E u is a finite union of such sets S and on each such set S, u is of the form stated in (2). This completes the proof of the theorem. Theorem 4.6. Let G be an arbitrary locally compact group and let u ∈ B(G) be such that E u is open in G. Then u is power bounded if and only if there exist n

(i) pairwise disjoint open sets F1 , . . . , F n in R(G) such that E u = ∪ F j and open j=1

subgroups H j of G and a j ∈ G such that F j ⊆ a j H j , j = 1, . . . , n, and (ii) characters γ j of H j and α j ∈ T , j = 1, . . . , n, such that u(x) = α j γ j (a−1 j x) for all

x ∈ Fj .

Theorem 4.6 was established in [12].

5 Some remarks and open problems A. Invariant complementation property Open problem 1. Does every locally compact group have the invariant complementation property?

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As is shown in [4] that if G is an amenable locally compact group, then G has the invariant complementation property. B. Natural projections. Let A be a commutative Banach algebra with a BAI. For f ∈ A∗ and a ∈ A, by a ⋅ f we denote the functional on A defined by ⟨a ⋅ f, b⟩ = ⟨f, ab⟩ . A projection P : A∗ → A∗ is said to be “invariant”(or A-invariant) if, for any a ∈ A and f ∈ A∗ , the equality P(a ⋅ f) = a ⋅ P(f) holds. Similarly, a closed subspace X of A∗ is said to be “invariant” if, for each a ∈ A and f ∈ X, the functional a ⋅ f is in X (i.e. X is an A-module for the natural action (a, f) 󳨃→ a ⋅ f). If there is an invariant projection from A∗ onto a closed invariant subspace X of A∗ then X is said to be “invariantly complemented in A∗ ”. We say that a projection P : A∗ 󳨃→ A∗ is “natural” if, for each γ ∈ ∆(A), either P(γ) = γ or P(γ) = 0. If X is a closed invariant subspace of A∗ and if there is natural projection P from A∗ onto X we shall say that X is “naturally complemented” in A∗ .

Lemma 5.1. Let P : A∗ → A∗ be a projection. Then (a) P is natural if and only if for each γ ∈ ∆(A) and a ∈ A, P(a ⋅ γ) = a ⋅ P(γ). (b) Every invariant projection P : A∗ → A∗ is natural.

Theorem 5.2. Let G be an amenable locally compact group, and I be a closed ideal in A(G). Then X = I ⊥ is invariantly complemented if and only if X is naturally complemented. C. Geometric form of the Hahn-Banach Separation Theorem The geometric form of Hahn-Banach Separation Theorem asserts that every closed vector subspace of a locally convex space is the intersection of the closed hyperplanes containing it. Lemma 5.3. Let H be a closed subgroup of G, and U be a neighbourhood basis of the identity of G. If G has the H-separation property, then (∗)

H = ⋂ HUH U∈U

Theorem 5.4 ([9]). If G is connected, then G has H-separation property if and only if (∗) holds.

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Lau

Open problem 2. If G has property (∗) for each closed subgroup of G, does G have the invariant complementation property? For general locally compact group G G − [SIN] ⇒G 󴀉󴁙󴁉

has separation 󳨐⇒

G has geometric separtion property

property ⇓ ⇑̸

Complementation property For connected G : G − [SIN] ⇐⇒ G

has separation ⇐⇒ G property

has geometric separtion property

D. Sets of spectral synthesis Let G be a locally compact group. To any closed subset E of G, the following two ideals are associated k(E) = {a ∈ A(G) : a = 0 on E} and j(E) = {a ∈ A(G) : The support of a is compact and disjoint from E}. The closed ideals J(E) = j(E) and k(E) are, respectively, the smallest and the largest closed ideals with hull E. When these two ideals coincide the set E is said to be a set of synthesis. A celebrated theorem due to Malliavin [19] states that every nondiscrete locally compact abelian group G contains a closed set that is not a set of synthesis for the algebra A(G). The same is true in the nonabelian case too [8]. Open problem 3. When is a set E in a locally compact group in R c (G) a set of synthesis? Acknowledgement. The author would like to thank the referee for his (or her) valuable suggestions.

Invariant Complementation Property

| 187

Bibliography [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10] [11] [12]

[13] [14]

[15] [16] [17] [18]

[19]

[20] [21] [22]

C.-H. Chu and A.T.-M. Lau, Jordan structures in harmonic functions and Fourier algebras on homogeneous spaces, Math. Ann. 336 no.4 (2006), 803-840. C.-H. Chu and A.T.-M. Lau, Harmonic functions on groups and Fourier, Lecture Notes in Mathematics 1782, Springer-Verlag, Berlin, 2002, pp. viii+100. . B.E. Forrest, Amenability and ideals in A(G), Austral. J. Math. Ser. A 53 (1992), 143-155. B.E. Forrest, E. Kaniuth, A.T.-M. Lau and N. Spronk, Ideals with bounded approximate identities in Fourier algebras, J. Funct. Anal. 203 (2003), 286-304. J.E. Gilbert,On projections of L∞ (G) and translation invariant subspaces, Proc. London Math. Soc. 19 (1969), 69-88. M. Ilie, N. Spronk, Completely bounded homomorphisms of the Fourier algebras, J. Funct. Anal. 225 (2005) 480-499. E. Kaniuth and A.T.-M. Lau, A separation property of positive definite functions on locally compact groups and applications to Fourier algebras, J. Funct. Anal. 175 no.1 (2000), 89110. E. Kaniuth and A.T.-M. Lau, Spectral synthesis for A(G) and subspaces of locally compact groups, Proc. of American Math. Society 129 (2001), 3253-3263. E. Kaniuth and A.T.-M. Lau, On a separation property of positive definite functions on locally compact groups, Math. Z. 243 no. 1 (2003), 161-177. E. Kaniuth and A.T.-M. Lau, Extensions and separation of positive definite functions on locally compact groups, Trans. Amer. Math. Soc. 359 no. 1 (2007), 447-463. E. Kaniuth, A.T.-M. Lau and A. Ülger. Multipliers of commutative Banach algebras, power boundedness and Fourier-Stieltjes algebras, J. London Math. Soc. 81 (2) (2010), 255-275. E. Kaniuth, A.T.-M. Lau and A. Ülger, Power boundedness in Fourier and Fourier Stieltjes algebras and other commutative Banach algebras, J. of Functional Analysis 260 (2011), 2191-2496. K. Kaniuth, A.T.-M. Lau and A. Ülger, Power boundedness in Banach algebras associated to locally compact groups, Studia Math. 222 no. 2 (2014), 165-189. E. Kaniuth and A.T.-M. Lau, Fourier and Fourier-Stieltjes Algebras on Locally Compact Groups,American Math. Society, Mathematical Surveys and Monographs 231 (2018), 306 pages. A.T.-M. Lau, Invariantly complemented subspaces of L∞ (G) and amenable locally compact groups, Illinois J. Math. 26 no 2 (1982), 226-235. A.T.-M. Lau and V. Losert, Weak∗ closed complemented invariant subspaces of L∞ (G) and amenable locally compact groups, Pacific J. Math. 123 no. 1 (1986), 149-159. A.T.-M. Lau and A. Ülger, Helson sets of synthesis are Ditkin sets, Proceedings A.M.S. 146 (2018), 203-206. A.T.-M. Lau and A. Ülger, Characterizations of closed ideals with bounded approximate identities in commutative Banach algebras, complemented subspaces of the group von Neumann algebras and applications, Transaction A.M.S. 366 (2014), 4151-4171. Paul Malliavin, Impossibilité de la synthèse spectrale sur les groupes abéliens non compacts, (French), 1959 Séminaire P. Lelong, Faculté des Sciences de Paris 178 1958/59, 178pp. H.P. Rosenthal, Projections onto translation-invariant subspaces of Lp(G), Memoir of Amer. Math. Soc. 63 (1966), 84pp. B. Schreiber, Measures with bounded convolution powers, Trans. Amer. Math. Soc. 151 (1970), 405-431. B. Schreiber, On the coset ring and strong Ditkin sets, Pacific J. Math. 33 (1970), 805-812.

188 |

[23]

Lau

M. Takesaki and N. Tatsuma, Duality and subgroups, Ann. of Math. 92 no. 2 (1971), 344364.

Niels Jakob Laustsen and Jared T. White

Subspaces that can and cannot be the kernel of a bounded operator on a Banach space Abstract: Given a Banach space E, we ask which closed subspaces may be realised as the kernel of a bounded operator E → E. We prove some positive results which imply in particular that when E is separable every closed subspace is a kernel. Moreover, we show that there exists a reflexive Banach space E which contains a closed subspace that cannot be realised as the kernel of any bounded operator on E. This implies that the Banach algebra B(E) of bounded operators on E fails to be weak*-topologically left Noetherian in the sense of [7]. The Banach space E that we use is the dual of one of Wark’s non-separable, reflexive Banach spaces with few operators. Keywords: Banach space, bounded operator, kernel, dual Banach algebra, weak*closed ideal, Noetherian. Classification: 46H10, 47L10; secondary: 16P40, 46B26, 47A05, 47L45.

1 Introduction In this note we address the following natural question: given a Banach space E, which of its closed linear subspaces F are the kernel of some bounded linear operator E → E? We shall begin by showing that if either E/F is separable, or F is separable and E has the separable complementation property, then F is indeed the kernel of some bounded operator on E (Propositions 2.1 and 2.2). Our main result is that there exists a reflexive, non-separable Banach space E for which these are the only closed linear subspaces that may be realised as kernels (Theorem 2.7), and in particular E has a closed linear subspace that cannot be realised as the kernel of a bounded linear operator on E (Corollary 2.8). The Banach space in question may be taken to be the dual of any reflexive, non-separable Banach space that has few operators, in the sense that every bounded operator on E is the sum of a scalar multiple of the identity and an operator with separable range. Wark has shown that such Banach spaces exist [5, 6]. Niels Jakob Laustsen , Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, United Kingdom. e-mail: [email protected] Jared T. White, Laboratoire de Mathématiques de Besançon, Université de FrancheComté, 16 Route de Gray, 25030 Besançon, France. e-mail: [email protected], [email protected] https://doi.org/10.1515/9783110602418-0011

190 | Laustsen-White We now describe how we came to consider this question. Given a Banach space E we write E󸀠 for its dual space, and B(E) for the algebra of bounded linear operators E → E. We recall that a dual Banach algebra is a Banach algebra A which is isomorphically a dual Banach space in such a way that the multiplication on A is separately weak*-continuous. When E is a reflexive Banach space, B(E) is ̂ E󸀠 , where ⊗ ̂ denotes the projective a dual Banach algebra with predual given by E⊗ tensor product of Banach spaces. We recall the following definition from [7]: Definition 1.1. Let A be a dual Banach algebra. We say that A is weak*-topologically left Noetherian if every weak*-closed left ideal I of A is weak*-topologically finitely-generated, i.e. there exists n ∈ N, and there exist x1 , . . . , x n ∈ I such that w∗ I = Ax1 + C x1 + ⋅ ⋅ ⋅ + Ax n + C x n . In [7] various examples were given of dual Banach algebras which satisfy this condition, but none were found that fail it. Using our main result, we are able to prove in Theorem 2.9 of this note that, for any non-separable, reflexive Banach space E with few operators in the above sense, B(E󸀠 ) is a dual Banach algebra which is not weak*-topologically left Noetherian.

2 Results We first show that in many cases closed linear subspaces can be realised as kernels. In particular, for a separable Banach space every closed linear subspace is the kernel of a bounded linear operator. Given a Banach space E, and elements x ∈ E, λ ∈ E󸀠 , we use the braket notation |x⟩⟨λ| to denote the rank-one operator y 󳨃→ ⟨y, λ⟩x. Proposition 2.1. Let E be a Banach space, and let F be a closed linear subspace of E such that E/F is separable. Then there exists T ∈ B(E) such that ker T = F.

Proof. Since E/F is separable, the unit ball of F ⊥ ≅ (E/F)󸀠 is weak*-metrisable, and hence, since it is also compact, it is separable. Therefore we may choose a sequence of functionals (λ n ) which is weak*-dense in the unit ball of F ⊥ . We may assume that E is infinite dimensional, since otherwise the result follows from elementary linear algebra. We may therefore pick a normalised basic sequence (b n ) in E. Define T ∈ B(E) by ∞

T = ∑ 2−n |b n ⟩⟨λ n |. n=1

Subspaces as Kernels of Bounded Operators |

191

Since each λ n belongs to F ⊥ , clearly F ⊂ ker T. Conversely, if x ∈ ker T then, since (b n ) is a basic sequence, we must have ⟨x, λ n ⟩ = 0 for all n ∈ N. Hence x ∈ {λ n : n ∈ N}⊥ = (spanw {λ n : n ∈ N})⊥ = (F ⊥ )⊥ = F, ∗

as required.

A Banach space E is said to have the separable complementation property if, for each separable linear subspace F of E, there is a separable, complemented linear subspace D of E such that F ⊂ D. For such Banach spaces we can show that every separable closed linear subspace is a kernel. By [2] every reflexive Banach space has the separable complementation property, so that the next proposition applies in particular to the duals of Wark’s Banach spaces, which we shall use in our main theorems. We refer to [1] for a survey of more general classes of Banach spaces that enjoy the separable complementation property. Proposition 2.2. Let E be a Banach space with the separable complementation property. Then, for every closed, separable linear subspace F of E, there exists T ∈ B(E) such that ker T = F.

Proof. Choose a separable, complemented linear subspace D of E such that F ⊂ D, and let P ∈ B(E) be a projection with range D. By Proposition 2.1, we can find S ∈ B(D) such that ker S = F. Then T : x 󳨃→ SPx + x − Px,

E → E,

defines a bounded linear operator on E. We shall now complete the proof by showing that ker T = F. Indeed, for each x ∈ ker T we have 0 = (idE −P)Tx = x − Px, so that Px = x. This implies that 0 = Tx = Sx, and therefore x ∈ F. Conversely, each x ∈ F satisfies Sx = 0 and Px = x, from which it follows that Tx = 0. We recall some notions from Banach space theory that we shall require. Let E be a Banach space. A biorthogonal system in E is a set {(x γ , λ γ ) : γ ∈ Γ} ⊂ E × E󸀠 , for some indexing set Γ, with the property that

{1 if α = β ⟨x α , λ β ⟩ = { 0 otherwise {

(α, β ∈ Γ).

A Markushevich basis for a Banach space E is a biorthogonal system

192 | Laustsen-White {(x γ , λ γ ) : γ ∈ Γ} in E such that {λ γ : γ ∈ Γ} separates the points of E and such that span{x γ : γ ∈ Γ} = E. For an in-depth discussion of Markushevich bases see [1], in which a Markushevich basis is referred to as an “M-basis”. We now prove a lemma and its corollary which we shall use to prove Corollary 2.8 below. Lemma 2.3. Let E be a Banach space containing an uncountable biorthogonal system. Then E contains a closed linear subspace F such that both F and E/F are nonseparable.

Proof. Let {(x γ , λ γ ) : γ ∈ Γ} be an uncountable biorthogonal system in E. We can write Γ = ⋃∞ n=1 Γ n , where Γ n = {γ ∈ Γ : ‖x γ ‖, ‖λ γ ‖ ≤ n}

(n ∈ N).

Since Γ is uncountable, there must exist an n ∈ N such that Γ n is uncountable. Let ∆ be an uncountable subset of Γ n such that Γ n \ ∆ is also uncountable, and set F = span{x γ : γ ∈ ∆}. The subspace F is non-separable since {x γ : γ ∈ ∆} is an uncountable set satisfying ‖x α − x β ‖ ≥

1 1 |⟨x α − x β , λ α ⟩| = n n

(α, β ∈ ∆, α ≠ β).

Let q : E → E/F denote the quotient map. It is well known that the dual map (E/F)󸀠 → E󸀠 is an isometry with image equal to F ⊥ . For each γ ∈ Γ n \ ∆ the functional λ γ clearly belongs to F ⊥ , so that there exists g γ ∈ (E/F)󸀠 such that q󸀠 (g γ ) = λ γ , and such that ‖g γ ‖ = ‖λ γ ‖. We now see that {q(x γ ) : γ ∈ Γ n \ ∆} is an uncountable 1/n-separated subset of E/F because q󸀠 :

1 1 |⟨q(x α ) − q(x β ), g α ⟩| = |⟨x α − x β , q󸀠 (g α )⟩| n n 1 1 = |⟨x α − x β , λ α ⟩| = (α, β ∈ Γ n \ ∆, α ≠ β). n n

‖q(x α ) − q(x β )‖ ≥

It follows that E/F is non-separable. Corollary 2.4. Let E be a non-separable, reflexive Banach space. Then E contains a closed linear subspace F such that both F and E/F are non-separable.

Proof. By [1, Theorem 5.1] E has a Markushevich basis {(x γ , λ γ ) : γ ∈ Γ}. The set Γ must be uncountable since E is non-separable and, by the definition of a Markushevich basis, span{x γ : γ ∈ Γ} = E. Hence the result follows from Lemma 2.3. We now move on to discuss our main example. Building on the work of Shelah ¯ [4], Wark constructed in [5] a reflexive Banach space E W with the and Steprans

Subspaces as Kernels of Bounded Operators

|

193

property that it is non-separable but has few operators in the sense that B(E W ) = C id E W + X(E W ),

(1)

where X(E W ) denotes the ideal of operators on E W with separable range. Recently Wark gave a second example of such a space with the additional property that the space is uniformly convex [6]. For the rest of our paper E W can be taken to be either of these spaces. In particular, the only properties of E W that we shall make use of are that it is reflexive, non-separable, and satisfies Equation (1). Remark 2.5. We briefly outline why the dual Banach algebra B(E󸀠W ) fits into the framework of [7]. A transfinite basis for a Banach space X is a linearly independent family {x α : α < γ} of vectors in X, where γ is some infinite ordinal, such that X0 = span{x α : α < γ} is dense in X, and with the property that there is a constant C ⩾ 1 such that, for each ordinal β < γ, the linear map P β : X0 → X defined by Pβ ( ∑ sα xα ) = ∑ sα xα α 0. Every element in the pointwise closure of B(V1 ), as a subset of functions on the weak-star compact space T ∗ (B(V2∗ )), can be viewed as a restriction of some a∗∗ ∈ V1∗∗ . Indeed, since T ∗ is bounded there exists a positive constant c > 0 such that T ∗ (B(V2∗ )) ⊂ cB(V1∗ ). Apply Lemma 4.5.3 to the inclusion map q : X1 = T ∗ (B(V2∗ )) 󳨅→ X2 = cB(V1∗ )

with X1 := T ∗ (B(V2∗ )) ⊂ X2 := cB(V1∗ ), F1 := B(V1 ), F2 := {v ∘ q : v ∈ B(V1 )}, where v ∘ q is a restriction of the map v : cB(V1∗ ) → R , ψ 󳨃→ ⟨v, ψ⟩ to T ∗ (B(V2∗ )). Now the equivalence (2) ⇔ (4) of Lemma 4.3 finishes the proof. (3) ⇔ (4) Lemma 4.5.4(a) (for the map q = T ∗ : B(V2∗ ) → T ∗ (B(V2∗ ))) implies that B(V1 ) is a Rosenthal family on T ∗ (B(V2∗ )) iff T(B(V1 )) is a Rosenthal family on B(V2∗ ). Now recall that Rosenthal family and tame family are the same in our setting by Lemma 4.3.

Tame functionals on Banach algebras

| 219

3 Tame functionals on the group algebra l1 (G) Let G be a discrete group. Denote by l1 (G) the usual (unital) Banach group algebra with respect to the convolution product ⋆ and the l1 -norm. For every s ∈ G we have δ s ∈ l1 (G), where δ s (s) = 1 and δ s (x) = 0 for every x ∈ G with x ≠ s. We have the embedding δ : G 󳨅→ l1 (G), s 󳨃→ δ s , where

δ s ⋆ δ t = δ st ∀s, t ∈ G.

(1)

For a group G and a real-valued function f : G → R the left translation fs by s ∈ G is defined as the function fs : G → R , (fs)(t) = f(st) and

fG := {fs : s ∈ G}.

Below the functional f ⋅ δ s for f ∈ l∞ (G) = l1 (G)∗ is defined as in Equation 1. Then (f ⋅ δ s )(t) = (f ⋅ δ s )(δ t ) = f(δ s ⋆ δ t ) = f(δ st ) = f(st).

This means that

f ⋅ δ s = fs, f ⋅ δ(G) = fG.

(2)

(3)

In contrast to the algebra L1 (G) for locally compact groups G, the algebra l1 (G) has the unit element δ e , where e is the unit element of G. Lemma 3.3 implies that span{δ(G)} is norm dense in l1 (G). Let l∞ (G) be the Banach space (usual sup-norm) of all bounded functions on G. For every continuous functional f : l1 (G) → R we have the corresponding restriction r(f) := f ∘δ : G → R. This defines the canonical isomorphism of Banach spaces r : l1 (G)∗ → l∞ (G). It reflects the standard fact that l1 (G)∗ can be identified with l∞ (G). For every s ∈ G define the functional σ s : l∞ → R , h 󳨃→ h(s). We have an injection

σ : G → l∗∞ (G), s 󳨃→ σ s

such that σ(G) is also discrete in the weak-star topology of l∗∞ . Furthermore, C(G) = l∞ (G) as a Banach space is naturally isometric to the Banach space C(βG),

220 |

Megrelishvili

where, βG is the Čech-Stone compactification of the discrete space G. Consider the natural topological embedding i : βG 󳨅→ l∞ (G)∗ = C(βG)∗ . Its restriction to G is just σ. So, the weak-star closure of σ(G) in l∞ (G)∗ can be naturally identified with βG.

3.1 Tame functions on groups The theory of tame dynamical systems developed in a series of works (see e.g. [5, 7, 8, 13, 17, 18]). Connections to other areas of mathematics like: Banach spaces, circularly ordered systems, substitutions and tilings, quasicrystals, cut and project schemes and even model theory and logic were established. See e.g. [1, 11, 12, 14] and the survey [9] for more details. Definition 3.1. (see for example [9, 12]) Let f : G → R be a bounded RUC (right uniformly continuous) function on a topological group G. Then f is said to be a tame function if fG := {fs : s ∈ G} is a tame family of functions on G; notation: f ∈ Tame(G). Tame functions on G are characterized in [8] as generalized matrix coefficients of isometric continuous representations of G on Rosenthal Banach spaces. Recall a similar result from [21] which asserts that weakly almost periodic functions on G are exactly matrix coefficients of representations on reflexive spaces. So, it is immediate from these results that WAP(G) ⊂ Tame(G). Note also that RUC(G) is exactly the set of all matrix coefficients for Banach representations of G. For more information about matrix coefficients of group representations we refer to [21] and [4]. There are many interesting tame functions on groups which are not WAP. For example, on the discrete integer group Z. Fibonacci bisequence c : Z → {0, 1} defined by Fibonacci substitution is tame but not WAP. It is well known that WAP(L1 (G)) = WAP(G) for every locally compact group G (see Lau [19] and Ülger [28]). In particular, for discrete G it implies that WAP(l1 (G)) = WAP(G). Our main result in this note, Theorem 3.2 below, shows that a similar result holds for the tame case. Theorem 3.2. Tame(l1 (G)) = Tame(G) for every discrete group G.

Proof. Tame(l1 (G)) ⊆ Tame(G). Let f ∈ Tame(l1 (G)). Then by Lemma 2.7 we know that the set Lf (B(l1 (G))) = f ⋅ B(l1 (G)) is a tame subset in l∞ (G). Since δ(G) ⊂ B(l1 (G)) we get that f ⋅ δ(G) ⊂ f ⋅ B(l1 (G)) is also a tame subset in l∞ (G). On the other hand, f ⋅ δ(G) = fG by Equation 3. Hence, fG is a tame subset as a family

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| 221

of functions on B(l∞ (G)∗ ). Therefore, fG is a tame subset also on the subset G = σ(G) ⊂ B(l∞ (G)∗ ). This means that f ∈ Tame(G). Tame(l1 (G)) ⊇ Tame(G). Let f ∈ Tame(G). Then fG is a tame family of functions on G. That is, fG = Lf (δ(G)) ⊂ l∞ (G)

is a tame family on σ(G) ⊂ l∞ (G)∗ = (l1 (G))∗∗ . Then by Lemma 4.4.3, fG is a tame family also on βG. Moreover, by Lemma 4.8 we obtain that fG is a tame family on the weak-star closed unit ball B(l∞ (G)∗ ) of l∞ (G)∗ = C(βG)∗ . Then the same is true for the union −fG ∪ fG. Now the convex hull co(−fG ∪ fG) is also tame family of functions on the compact space B(l∞ (G)∗ ) by Lemma 4.7. Lemma 3.3. W := co(−δ(G) ∪ δ(G)) is norm dense in the unit ball of l1 (G).

Proof. Let v ∈ l1 (G) with ||v|| ≤ 1. Then, by definition, v is a function v : G → R such that the support supp(v) is at most countable, v = ∑s∈supp(v) v(s)δ s . It is common to write it simply v = ∑ c s δ s , where c s = v(s). Then ||v|| = ∑ |c s | ≤ 1. For a given ε > 0 we need to find w ∈ W such that ||v − w|| < ε. There exists a finite subset J ⊂ supp(v) such that ||v − Σ s∈J c s δ s || < ε. Define w := Σ s∈J c s δ s . Now observe that w ∈ W. Indeed, we may suppose that J = J+ ∪ J− is the disjoint union where J+ correspond to positive coefficients and J− to negative coefficients. Then w = ∑ c s δ s = ∑ c s δ s + ∑ (−c s )(−δ s ) + s∈J

where

∆ ∆ δ e + (−δ e ) ∈ W, 2 2

s∈J−

s∈J+

∆ := 1 − ∑ c s + ∑ (−c s ) = 1 − ∑ |c s | ≥ 0. s∈J+

s∈J

s∈J−

We claim that the weak closure f ⋅ W of f ⋅ W in l∞ (G) contains f ⋅ B(l1 (G)). Since Lf is a linear operator, taking into account f ⋅ δ s = fs, we have w

f ⋅ W = f ⋅ co(−δ(G) ∪ δ(G)) = co(f ⋅ (−δ(G) ∪ δ(G)) = co(−fG ∪ fG).

Lemma 3.3 means that W

||⋅||

= B(l1 (G)). Since the operator Lf : l1 (G) → l∞ (G)

is norm continuous, the norm closure f ⋅ W Summing up we get

||⋅||

f ⋅ W = co(−fG ∪ fG) = co(−fG ∪ fG) w

w

||⋅||

of Lf (W) = f ⋅ W contains f ⋅ W

=f⋅W

||⋅||

⊇f⋅W

||⋅||

= f ⋅ B(l1 (G)).

||⋅||

.

Now we use Lemma 2.2 which implies that the weak closure co(−fG ∪ fG) , hence also its subfamily f ⋅ B(l1 (G)), are tame families on the weak-star compact ball B(l∞ (G))∗ . Finally Lemma 2.7 finishes the proof. w

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Remark 3.4. One may prove similarly the following formulas: (1) WAP(l1 (G)) = WAP(G). (2) Asp(l1 (G)) = Asp(G). The second formula is new. In the first case we may use Grothendieck’s Double Limit Property characterization of weakly precompact subsets. In the second case – properties of fragmentable families. Namely, one may show that T is Asplund operator iff T(B(V1 )) is a fragmented family on B(V2∗ ). The proof is similar to the proof of Lemma 2.7. Remark 3.5. The sets WAP(G), Asp(G), Tame(G) are distinct even for the discrete group Z. Indeed, the Fibonacci bisequence c : Z → {0, 1} is a tame function on Z but not Asplund (see [11, 12]). The characteristic function χ N : Z → {0, 1} is Asplund but not WAP. Hence, the sets of functionals WAP(l1 (Z)), Asp(l1 (Z)), Tame(l1 (Z)) are also distinct. We are going to investigate tame functionals in some future works. Among others we intend to deal with the following natural question. Question 3.6. Is it true that Tame(L1 (G)) = Tame(G) for every (nondiscrete) locally compact group G?

4 Appendix 4.1 Background on fragmentability and tame families The following definitions provide natural generalizations of the fragmentability concept [16]. Definition 4.1. Let (X, τ) be a topological space and (Y, μ) a uniform space. (1) [15, 20] X is (τ, μ)-fragmented by a (typically, not continuous) function f : X → Y if for every nonempty subset A of X and every ε ∈ μ there exists an open subset O of X such that O ∩ A is nonempty and the set f(O ∩ A) is εsmall in Y. We also say in that case that the function f is fragmented. Notation: f ∈ F(X, Y), whenever the uniformity μ is understood. If Y = R then we write simply F(X). (2) [7] We say that a family of functions F = {f : (X, τ) → (Y, μ)} is fragmented if condition (1) holds simultaneously for all f ∈ F. That is, f(O ∩ A) is ε-small for every f ∈ F. (3) [8] We say that F is an eventually fragmented family if every infinite subfamily C ⊂ F contains an infinite fragmented subfamily K ⊂ C.

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In Definition 4.1.1 when Y = X, f = id X and μ is a metric uniformity, we retrieve the usual definition of fragmentability (more precisely, (τ, μ)-fragmentability) in the sense of Jayne and Rogers [16]. Implicitly it already appears in a paper of Namioka and Phelps [24]. Lemma 4.2. [7, 8] (1) If f : (X, τ) → (Y, μ) has a point of continuity property PCP (i.e., for every closed nonempty A ⊂ X the restriction f|A : A → Y has a continuity point) then it is fragmented. If (X, τ) is hereditarily Baire (e.g., compact, or Polish) and (Y, μ) is a pseudometrizable uniform space then f is fragmented if and only if f has PCP. So, in particular, for compact X, the set F(X) is exactly B󸀠r (X) in the notation of [27]. (2) If X is Polish and Y is a separable metric space then f : X → Y is fragmented iff f is a Baire class 1 function (i.e., the inverse image of every open set is F σ ). For other properties of fragmented maps and fragmented families we refer to [7, 8, 10, 15, 20, 21, 23]. Basic properties and applications of fragmentability in topological dynamics can be found in [8, 9, 10].

4.2 Independent sequences of functions The following useful theorem synthesizes some known results. It is mainly based on results of Rosenthal and Talagrand. The equivalence of (1), (3) and (4) is a part of [27, Theorem 14.1.7] For the case (1) ⇔ (2) note that every bounded independent sequence {f n : X → R}n∈N is an l1 -sequence (in the sup-norm), [25, Prop. 4]. On the other hand, as the proof of [25, Theorem 1] shows, if {f n }n∈N has no independent subsequence then it has a pointwise convergent subsequence. Bounded pointwise-Cauchy sequences in C(X) (for compact X) are weak-Cauchy as it follows by Lebesgue’s theorem. Now Rosenthal’s dichotomy theorem [25, Main Theorem] asserts that {f n } has no l1 -sequence. In [8, Sect. 4] we show why eventual fragmentability of F can be included in this list (item (5)). Lemma 4.3. Let X be a compact space and F ⊂ C(X) a bounded subset. The following conditions are equivalent: (1) F does not contain an l1 -sequence. (2) F is a tame family (does not contain an independent sequence). (3) Each sequence in F has a pointwise convergent subsequence in R X . (4) F is a Rosenthal family (meaning that the pointwise closure cl (F) of F in R X consists of fragmented maps; that is, cl (F) ⊂ F(X)). (5) F is an eventually fragmented family.

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Lemma 4.4. (1) Let q : X1 → X2 be a map between sets and {f n : X2 → R}n∈N a bounded sequence of functions (with no continuity assumptions on q and f n ). If {f n ∘ q} is an independent sequence on X1 then {f n } is an independent sequence on X2 . (2) If q is onto then the converse is also true. That is {f n ∘ q} is independent if and only if {f n } is independent. (3) Let {f n } be a bounded sequence of continuous functions on a topological space X. Let Y be a dense subset of X. Then {f n } is an independent sequence on X if and only if the sequence of restrictions {f n |Y } is an independent sequence on Y. Proof. Claims (1) and (2) are straightforward. (3) Since {f n } is an independent sequence for every pair of finite disjoint sets P, M ⊂ N, the set ⋂ f n−1 (−∞, a) ∩ ⋂ f n−1 (b, ∞) n∈P

n∈M

is non-empty. This set is open because every f n is continuous. Hence, each of them −1 meets the dense set Y. As f n−1 (−∞, a) ∩ Y = f n |−1 Y (−∞, a) and f n (b, ∞) ∩ Y = −1 f n |Y (b, ∞), this implies that {f n |Y } is an independent sequence on Y. Conversely if {f n |Y } is an independent sequence on a subset Y ⊂ X then by (1) (where q is the embedding Y 󳨅→ X), {f n } is an independent sequence on X.

Lemma 4.5. Let q : X1 → X2 be a map between sets. (1) The natural map γ : R X2 → R X1 , γ(ϕ) = ϕ ∘ q is pointwise continuous. (2) If q : X1 → X2 is onto then γ is injective. (3) F2 ⊂ [a, b]X2 and F1 ⊂ [a, b]X2 be subsets such that F1 = F2 ∘ q. Then γ(cl p (F2 )) = cl p (F1 ). (4) Let q : X1 → X2 be a continuous onto map between compact spaces, F2 ⊂ C(X2 ) and F1 ⊂ C(X1 ) be norm bounded subsets such that F1 = F2 ∘ q. Then (a) F1 is a Rosenthal family for X1 if and only if F2 is a Rosenthal family for X2 . (b) γ induces a homeomorphism between the compact spaces cl p (F2 ) and cl p (F1 ).

Proof. Claims (1) and (2) are trivial. (3)(a): By the continuity of γ we get γ(F2 ) ⊂ γ(cl p (F2 )) ⊂ cl p (γ(F2 )). Since F2 is bounded the set cl p (F2 ) is compact in R X2 . Then γ(cl p (F2 )) = cl p (γ(F2 )). On the other hand, γ(F2 ) = F2 ∘ q = F1 . Therefore, cl p (F2 ) ∘ q = cl p (F1 ). (4)(a) Use (3) and Lemma 4.6. (3)(b): Combine the assertions (1) and (2) taking into account that γ(cl p (F2 )) = cl p (γ(F2 )) = cl p (F1 ). Recall some useful Lemmas from [8].

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Lemma 4.6. Let α : X → X 󸀠 be a continuous onto map between compact spaces. Assume that (Y, μ) is a uniform space, f : X → Y and f 󸀠 : X 󸀠 → Y are maps such that f 󸀠 ∘ α = f . Then f is a fragmented map if and only if f 󸀠 is a fragmented map.

Lemma 4.7. Let F ⊂ C(X) be a tame (eq. Rosenthal) family for a compact space X. Then its convex hull co(F) is also a tame family for X.

Lemma 4.8. Let X be a compact space and F ⊂ C(X) be a bounded family. Then F is a tame family for X if and only if F is a tame family for the weak-star compact unit ball B(C(X)∗ ).

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14] [15]

J. B. Aujogue, Ellis enveloping semigroup for almost canonical model sets of an Euclidean space, Algebr. Geom. Topol. 15 (2015), no. 4, 2195–2237. W.J. Davis, T. Figiel, W.B. Johnson and A. Pelczyński, Factoring weakly compact operators, J. of Funct. Anal. 17 (1974), 311–327. M. Filali, M. Neufang and M.S. Monfared, Representations of Banach algebras subordinate to topologically introverted spaces, Trans. Amer. Math. Soc. 367 (2015), 8033–8050. J. Galindo, On Group and Semigroup Compactifications of Topological Groups, Preprint. http://www3.uji.es/ ∼ jgalindo/Inv/notesSemCompBook.pdf E. Glasner, On tame dynamical systems, Colloq. Math. 105 (2006), 283–295. E. Glasner, The structure of tame minimal dynamical systems, Ergod. Th. and Dynam. Sys. 27, (2007), 1819–1837. E. Glasner and M. Megrelishvili, Linear representations of hereditarily non-sensitive dynamical systems, Colloq. Math. 104 (2006), no. 2, 223–283. E. Glasner and M. Megrelishvili, Representations of dynamical systems on Banach spaces not containing l1 , Trans. Amer. Math. Soc. 364 (2012), 6395–6424. E. Glasner and M. Megrelishvili, Representations of dynamical systems on Banach spaces, Recent Progress in General Topology III, Springer-Verlag, Atlantis Press, 2014. E. Glasner and M. Megrelishvili, Eventual nonsensitivity and tame dynamical systems, arXiv:1405.2588, 2014. E. Glasner and M. Megrelishvili, Circularly ordered dynamical systems, Monats. Math. 185 (2018), 415–441. E. Glasner and M. Megrelishvili, More on tame dynamical systems, in: Lecture Notes S. vol. 2013, Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, Eds.: S. Ferenczi, J. Kulaga-Przymus, M. Lemanczyk, Springer, 2018. E. Glasner, M. Megrelishvili and V.V. Uspenskij, On metrizable enveloping semigroups, Israel J. of Math. 164 (2008), 317–332. T. Ibarlucia, The dynamical hierachy for Roelcke precompact Polish groups, Israel J. Math. 215 (2016), 965–1009. J.E. Jayne, J. Orihuela, A.J. Pallares and G. Vera, σ-fragmentability of multivalued maps and selection theorems, J. Funct. Anal. 117 (1993), no. 2, 243–273.

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[16] J.E. Jayne and C.A. Rogers, Borel selectors for upper semicontinuous set-valued maps, Acta Math. 155 (1985), 41–79. [17] D. Kerr and H. Li, Independence in topological and C ∗ -dynamics, Math. Ann. 338 (2007), 869–926. [18] A. Köhler, Enveloping semigrops for flows, Proc. of the Royal Irish Academy 95A (1995), 179–191. [19] A. T. Lau, Closed convex invariant subsets of L p (G), Trans. Amer. Math. Soc. 232 (1977), 131–142. [20] M. Megrelishvili, Fragmentability and continuity of semigroup actions, Semigroup Forum 57 (1998), 101–126. [21] M. Megrelishvili, Fragmentability and representations of flows, Topology Proceedings, series 27:2 (2003), 497–544. See also: www.math.biu.ac.il/ ̃megereli. [22] M. Megrelishvili, A note on tameness of families having bounded variation, Topology and Appl. 217 (2017), 20–30. [23] I. Namioka, Radon-Nikodým compact spaces and fragmentability, Mathematika series 34 (1987), 258–281. [24] I. Namioka and R.R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), 735–750. [25] H.P. Rosenthal, A characterization of Banach spaces containing l1 , Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. [26] E. Saab and P. Saab, A dual geometric characterization of Bancah spaces not containing l1 , Pacific J. Math. 105:2 (1983), 413–425. [27] M. Talagrand, Pettis integral and measure theory, Memoirs of AMS No. 51 (1984). [28] A. Ülger, Continuity of weakly almost periodic functionals on L p (G), Quart. J. Math. Oxford (2) 37 (1986), 495–497. [29] N.J. Young, Periodicity of functionals and representations of normed algebras on reflexive spaces, Proc. Edinburgh Math. Soc. 20 (1976), 99–120.

Jared T. White

Left Ideals of Banach Algebras and Dual Banach Algebras Abstract: We investigate topologically left Noetherian Banach algebras. We show that if G is a compact group, then L 1 (G) is topologically left Noetherian if and only if G is metrisable. We prove that, given a Banach space E such that E󸀠 has BAP, the algebra of compact operators K(E) is topologically left Noetherian if and only if E󸀠 is separable; it is topologically right Noetherian if and only if E is separable. We then give some examples of dual Banach algebras which are topologically left Noetherian in the weak*-topology. Finally we give a unified approach to classifying the weak*-closed left ideals of certain dual Banach algebras that are also multiplier algebras, with applications to M(G) for G a compact group, and B(E) for E a reflexive Banach space with AP. Keywords: Noetherian, Banach algebra, dual Banach algebra, Multiplier algebra, weak*-closed ideal. Classification: Primary: 16P40, 46H10; secondary: 43A10, 43A20, 47L10.

1 Introduction Those studying Banach algebras have long been interested in the interplay between abstract algebra and abstract analysis. This motivates the comparison of the following two definitions: Definition 1.1. Let A be a Banach algebra, and let I be a closed left ideal of A. (i) Given n ∈ N, we say that I is (algebraically) generated by x1 , . . . , x n ∈ I if I = A♯ x1 + ⋅ ⋅ ⋅ + A♯ x n .

When there exist such elements x1 , . . . , x n for some n ∈ N we say that I is (algebraically) finitely-generated. (ii) Given n ∈ N, we say that I is topologically generated by x1 , . . . , x n ∈ I if I = A♯ x1 + ⋅ ⋅ ⋅ + A♯ x n .

When there exist such elements x1 , . . . , x n for some n ∈ N we say that I is topologically finitely-generated. Jared T. White, Laboratoire de Mathématiques de Besançon, Université de FrancheComté, 16 Route de Gray, 25030 Besançon, France, e-mail: [email protected], [email protected] https://doi.org/10.1515/9783110602418-0014

228 | White Here A♯ denotes the unitisation of A. It seems both natural and obvious that Definition 1.1(ii) is the appropriate one for the normed setting since it takes account of the topology. However, as Banach algebraists we wish to establish a more precise picture of exactly how these two definitions play out. One result in this spirit is the beautiful theorem of Sinclair and Tullo from 1974 [32]: Theorem 1.2. Let A be a Banach algebra which is (algebraically) left Noetherian. Then A is finite-dimensional. In recent years a number of papers have appeared which further illustrate that algebraic finite-generation of left ideals in Banach algebras is a very strong condition. This has been motivated by a conjecture of Dales and Żelazko [12] which states that a unital Banach algebra in which every maximal left ideal is finitelygenerated is finite-dimensional. The conjecture is known to hold for many classes of Banach algebras [4, 11, 12, 36]. For example, in [36] the present author verified the conjecture for many of the algebras coming from abstract harmonic analysis, including the measure algebra of a locally compact group, as well as a large class of Beurling algebras. In this article we aim to fill in another corner of this picture and contrast with the above results by investigating topologically left Noetherian Banach algebras, which we define as follows: Definition 1.3. Let A be a Banach algebra. We say that A is topologically left Noetherian if every closed left ideal is topologically finitely-generated. We shall demonstrate below that, in contrast to the Sinclair–Tullo Theorem, there are many infinite-dimensional examples of topologically left Noetherian Banach algebras, and that, moreover, the condition often picks out a nice property of some underlying group or Banach space. We note that versions of Noetherianity for topological rings and algebras are not at all new, and various different versions of this notion exist: see e.g. [7, 27]. Moreover, topological left Noetherianity as in Definition 1.3 was the subject of a post by Kevin Casto on Mathoverflow [6]. His question is partially answered by our Theorem 1.6. We also mention that the fact that separable C*-algebras are topologically left Noetherian has been known since Prosser’s 1963 memoir [30], in which it was shown that every closed left ideal of a separable C*-algebra is topologically principal [30, Corollary, pg. 26]. It should be noted that using modern techniques a short proof of this fact can be obtained by using the correspondence between the closed left ideals of a C*-algebra and its hereditary C*-subalgebras, and the fact that, in the separable case, hereditary C*-subalgebras of a C*-algebra A are all of the form xAx, for some positive element x ∈ A.

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One natural condition that we could consider for Banach algebras that is different from ours would be an ascending chain condition on chains of closed left ideals as in [7, Proposition 4.1]. However, we know of no infinite-dimensional examples of Banach algebras satisfying this condition. Indeed, many of the natural examples of Banach algebras satisfying Definition 1.3 are easily seen to fail the ascending chain condition. These include C[0, 1], as well as the infinitedimensional examples in Theorems 1.4 and 1.5 below. One might be tempted to try to prove that there are no infinite-dimensional Banach algebras satisfying an ascending chain condition for closed left ideals. However, if this were true, it would imply a negative solution to the question of whether there exists an infinite-dimensional, topologically simple, commutative Banach algebra, which is a notorious open question. For these reasons we have chosen only to study Definition 1.3 in this article. We now outline the main results of the paper. In Section 2 we shall fix some notation and prove some general results that we shall require in the sequel. Section 3 is concerned with Banach algebras on groups, and our main result is the following: Theorem 1.4. Let G be a compact group. Then L 1 (G) is topologically left Noetherian if and only if G is metrisable. We also prove an analogous result for the Fourier algebra of certain discrete groups (Proposition 3.1). In Section 4 we turn our attention to algebras of operators on a Banach space. We denote the set of compact operators on a Banach space E by K(E). Our main result is the following: Theorem 1.5. (i) Let E be a Banach space with AP. Then K(E) is topologically left Noetherian if and only if E󸀠 is separable. (ii) Let E be a Banach space such that E󸀠 has BAP. Then K(E) is topologically right Noetherian if and only if E is separable. This theorem actually follows from more general results about the algebra of approximable operators A(E), for a formally larger class of Banach spaces E (Theorem 4.5 and Theorem 4.9). In Section 5 we consider weak*-topologically left Noetherian dual Banach algebras, believing that this will be a more appropriate notion for the measure algebra M(G) of a locally compact group G, and for B(E), the algebra of bounded linear operators on a reflexive Banach space E. We consider Banach algebras A for which the multiplier algebra M(A) is a dual Banach algebra in a natural way, and prove a general result, Proposition 5.3, which says that M(A) is weak*topologically left Noetherian whenever A is topologically left Noetherian. We then

230 | White apply this theorem to the algebras of the form M(G) and B(E) to get the following corollary: Corollary 1.6. (i) Let G be a compact, metrisable group. Then M(G) is weak*-topologically left Noetherian. (ii) Let E be a separable, reflexive Banach space with AP. Then B(E) is weak*topologically left and right Noetherian. We then consider a more restricted class of Banach algebras, and we formulate an abstract approach for relating the ideal structure of a Banach algebra A belonging to this class to the weak*-ideal structure of M(A) (Theorem 5.10). In Proposition 5.11 we use this to show that, for this class, weak*-topological left Noetherianity of M(A) is equivalent to a ‖⋅‖-topological condition on A. In Section 6 we demonstrate how Theorem 5.10 gives a unified strategy for classifying the weak*-closed left ideals of both M(G), for G a compact group, and B(E), for E a reflexive Banach space with AP. We then observe that this leads to classifications of the closed right submodules of the predules. Finally, we mention that in [25] the author of the present work together with Niels Laustsen will show that there is a certain reflexive Banach space E which has AP such that B(E) fails to be weak*-topologically left Noetherian. This is the only example of a dual Banach algebra that we know of that fails to have this property.

2 Preliminaries We first fix some general notation. Given a locally compact group G, we denote by L 1 (G) the Banach algebra of integrable functions on G, which we refer to as the group algebra of G. We write M(G) for the Banach algebra of complex, regular Borel measures on G, known here as the measure algebra. We write A(G) for the Fourier algebra of G, as defined by Eymard in [13]. We write C(G) for the linear space of complex-valued continuous functions on G, and C0 (G) for the subspace consisting of functions vanishing at infinity. Of course, C0 (G) is a Banach space with the supremum norm; when G is compact C(G) = C0 (G), and we prefer the former notation over the latter. By a representation of G we implicitly mean a continuous unitary representation of G on a Hilbert space. We write ̂ G for the unitary dual of G, and a typical elê ment of G will be represented as (π, H π ), where H π is a Hilbert space, and π is an irreducible representation of G on H π . Given an arbitrary representation (π, H π ) of G, and vectors ξ, η ∈ H π , we write ξ ∗π η for the function on G defined by

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t 󳨃→ ⟨π(t)ξ, η⟩ (t ∈ G). We shall denote the modular function on G by ∆. We also use the notation f ̌(t) = f(t−1 ) (t ∈ G) for f ∈ L 1 (G). For a Banach algebra A we denote by A♯ the (conditional) unitisation of A, and by M(A) the multiplier algebra of A. We write CLI(A) for the lattice of closed left ideals of A, and CRI(A) for the lattice of closed right ideals. We denote the left action of A on its dual by a ⋅ λ for a ∈ A and λ ∈ A󸀠 , and set A ⋅ A󸀠 = {a ⋅ λ : a ∈ A, λ ∈ A󸀠 }.

Usually A will have a bounded approximate identity, in which case, by Cohen’s Factorisation Theorem, A ⋅ A󸀠 = span(A ⋅ A󸀠 ). We use similar notation for the right action. Given a Banach space E, we shall denote its dual space by E󸀠 . We write B(E) for the algebra of bounded linear operators E → E. We write K(E) for the ideal of compact operators, A(E) for the approximable operators, and F(E) for the finite rank operators. We write SUB(E) for the lattice of closed linear subspaces of E. Given x ∈ E and λ ∈ E󸀠 , we write x ⊗ λ for the rank one operator y 󳨃→ λ(y)x (y ∈ E). Given subsets X ⊂ E and Y ⊂ E󸀠 , we use the notation X ⊥ = {λ ∈ E󸀠 : ⟨x, λ⟩ = 0 (x ∈ X)},

Y⊥ = {u ∈ E : ⟨u, φ⟩ = 0 (φ ∈ Y)},

and we recall the well-known formulae (X ⊥ )⊥ = span(X),

(Y⊥ )⊥ = spanw (Y). ∗

(1)

A Banach space E is said to have the approximation property, or simply AP, if, whenever F is another Banach space, we have A(F, E) = K(F, E). There is also an equivalent formulation of the approximation property which has some useful generalizations: a Banach space E has AP if and only if, for every compact subset K ⊂ E and every ε > 0, there exists T ∈ F(E) such that ‖Tx − x‖ < ε (x ∈ K) [26, Theorem 3.4.32]. We say that E has the bounded approximation property, or BAP, if there exists a constant C > 0 such that the operator T can be chosen to have norm at most C. Clearly BAP implies AP. Moreover, a reflexive Banach space with AP has BAP [5, Theorem 3.7]. Many Banach spaces have the bounded approximation property: for instance any Banach space with a Schauder basis [26, Theorem 4.1.33] has BAP, and it can be deduced from this that any Hilbert space has BAP. The Banach space B(H), for H an infinite-dimensional Hilbert space, does not even have AP [34]. We recall from [31] that a dual Banach algebra is a Banach algebra A which is isomorphically a dual Banach space, in such a way that the multiplication is separately weak*-continuous. Examples include the measure algebra M(G) of a locally compact group G, with predual C0 (G), as well as B(E) for a reflexive Banach space

232 | White ̂ E󸀠 , where ⊗ ̂ denotes the projective tensor product of E, with predual given by E⊗ Banach spaces. Recall that a semi-topological algebra is a pair (A, τ), where A is an algebra, and τ is a topology on A such that (A, +, τ) is a topological vector space, and such that multiplication on A is separately continuous. For example, a dual Banach algebra with its weak*-topology is a semi-topological algebra. Let (A, τ) be a semi-topological algebra. Let I be a closed left ideal of A, and let n ∈ N. We say that I is τ-topologically generated by elements x1 , . . . , x n ∈ I if I = A♯ x1 + ⋅ ⋅ ⋅ + A♯ x n .

We say that I is τ-topologically finitely-generated if there exist n ∈ N and x1 , . . . , x n ∈ I which τ-topologically generate I. We say that I is τ-topologically principal if I is of the form A♯ x, for some x ∈ I. We say that A is τ-topologically left Noetherian if every closed left ideal of A is τ-topologically finitely-generated. For example, we shall often discuss weak*-topologically left Noetherian dual Banach algebras. When the topology is the norm topology on a Banach algebra we may simply speak of “topologically finitely-generated left ideals” et cetera. Analogously we may define τ-topologically finitely-generated right ideals, as well as τ-topologically right Noetherian algebras. If the algebra in question is commutative we usually drop the words “left” and “right”. We note that when a semi-topological algebra A has a left approximate identity we have A♯ x1 + ⋅ ⋅ ⋅ + A♯ x n = Ax1 + ⋅ ⋅ ⋅ + Ax n , for each n ∈ N , and each x1 , . . . , x n ∈ A. When this is the case we usually drop the unitisations in order to ease notation. For example, in the proof of Theorem 3.4 below, we shall write L 1 (G) ∗ g in place of L 1 (G)♯ ∗ g, for G a locally compact group and g ∈ L 1 (G). The following lemma will be invaluable throughout this article.

Lemma 2.1. Let A be a semi-topological algebra with a left approximate identity. Let J be a dense right ideal of A. Then J intersects every closed left ideal of A densely. Proof. Let (e α ) be a left approximate identity for A, which we may assume belongs to J (if not, then for each open neighbourhood of the origin U, and each index α, choose f α,U ∈ J such that e α − f α,U ∈ U. Then (f α,U ) is easily seen to be a left approximate identity for A.) Let I be a closed left ideal of A and let a ∈ I. Then for every index α we have e α a ∈ J ∩ I. Since a = limα e α a ∈ J ∩ I, and a was arbitrary, it follows that J ∩ I = I, as required. Next we show that τ-topological left Noetherianity is stable under taking quotients and extensions. For this lemma only we drop the τs and write “topologically

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left Noetherian” et cetera, even though we are not necessarily talking about a topology induced by a norm. Lemma 2.2. Let A be a semi-topological algebra, and let I be a closed (two-sided) ideal of A. (i) If A is topologically left Noetherian then so is A/I. (ii) Suppose that both I and A/I are topologically left Noetherian. Then so is A. (iii) A is topologically left Noetherian if and only if A♯ is topologically left Noetherian. Proof. Parts (i) and (ii) follow from routine arguments. For part (iii) we may suppose that A is non-unital for otherwise the result is trivial. If A is topologically left Noetherian then, since A♯ /A ≅ C is topologically left Noetherian, it follows from (ii) that A♯ is also. The converse follows from the fact that every closed left ideal of A is also a closed left ideal of A♯ .

3 Examples From Abstract Harmonic Analysis In this section we shall prove Theorem 1.4. It is surely easiest to determine whether or not a Banach algebra is topologically left Noetherian when we know what its closed left ideals are. Fortunately, this is the case for the group algebra of a compact group, as well as for the Fourier algebra of certain discrete groups, including all amenable groups. As a sort of warm up for the proof of Theorem 1.4 we shall show that, for such groups, the Fourier algebra A(G) is topologically Noetherian if and only if G is countable. Both proofs involve similar ideas. Proposition 3.1. Let G be a discrete group such that f ∈ A(G)f for all f ∈ A(G). Then A(G) is topologically Noetherian if and only if G is countable. Proof. Given E ⊂ G, write I(E) = {f ∈ A(G) : f(x) = 0, x ∈ E}. By [23, Proposition 2.2] the closed ideals of A(G) are all of the form I(E) for some subset E of G. Suppose first that G is countable and let I ⊲ A(G) be closed. Let E ⊂ G be such 1 that I = I(E), and enumerate G \ E = {x1 , x2 , . . . , }. Define g = ∑∞ n=1 n2 δ x n ∈ A(G). It is clear that supp g = G \ E, and hence that {x ∈ G : f(x) = 0 for every f ∈ A(G)♯ g} = E.

It follows from the classification of the closed ideals of A(G) given above that I = A(G)♯ g. As I was arbitrary we conclude that A(G) is topologically Noetherian. Now suppose that A(G) is topologically Noetherian. Then there exist n ∈ N and h1 , . . . , h n ∈ A(G) such that A(G) = A(G)♯ h1 + ⋅ ⋅ ⋅ + A(G)♯ h n . Since

234 | White A(G) ⊂ c0 (G), every function in A(G) must have countable support. Hence S := ⋃ni=1 supp h i is a countable set. Every f ∈ A(G)♯ h1 + ⋅ ⋅ ⋅ + A(G)♯ h n has supp f ⊂ S, and of course, after taking closures, we see that this must hold for every f ∈ A(G). This clearly forces S = G, so that G must be countable. Remark. The hypothesis of the previous proposition is satisfied by any discrete, amenable group, since in this case A(G) has a bounded approximate identity by Leptin’s Theorem, as well as many other groups including the free group on n generators for each n ∈ N [34]. The question of whether there are any locally compact groups which do not satisfy f ∈ A(G)f for every f ∈ A(G) is considered a difficult open problem. We now recall some facts about compact groups. Firstly, for G a compact group the closed left ideals of L 1 (G) have the following characterisation [21, Theorem 38.13]: Theorem 3.2. Let G be a compact group, and let I be a closed left ideal of L 1 (G). Then there exist linear subspaces E π ⊂ H π (π ∈ ̂ G) such that I = {f ∈ L 1 (G) : π(f)(E π ) = 0, π ∈ ̂ G} .

Let G be a compact group. Given π ∈ ̂ G we write T π (G) = span{ξ ∗π η : ξ, η ∈ H π }, and we write T(G) = span{ξ ∗π η : ξ, η ∈ H π , π ∈ ̂ G}. We recall the following facts about these spaces from [20, 21]: Theorem 3.3. Let G be a compact group. (i) Let σ, π ∈ ̂ G with σ ≠ π. Then σ(ξ ∗π η) = 0. (ii) The linear space T(G) is a dense ideal in L 1 (G). (iii) For each π ∈ ̂ G the space T π (G) is an ideal in L 1 (G), and as an algebra T π (G) ≅ M d π (C), where d π denotes the dimension of H π . Proof. The formula ⟨σ(f)ζ1 , ζ2 ⟩ = ∫ f(t)⟨σ(t)ζ1 , ζ2 ⟩ dt, G

for f ∈ L 1 (G), σ ∈ ̂ G and ζ1 , ζ2 ∈ H σ is well-known. Part (i) follows from this and the orthogonality relations [21, Theorem 27.20 (iii)]. Part (ii) follows from [21, Theorem 27.20, Lemma 31.4], and part (iii) follows from [21, Theorem 27.21]. We now prove the main theorem of this section, Theorem 3.4. Observe that Theorem 1.4 is simply “(a) if and only if (c)”. The equivalence of conditions (b) and (c) has surely been noticed before, but we include the proof to make our argument more transparent.

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Theorem 3.4. Let G be a compact group. Then the following are equivalent: (a) L 1 (G) is topologically left Noetherian; (b) ̂ G is countable; (c) G is metrisable. Proof. We first demonstrate that (b) implies (c). Our method is to show that G is first-countable, which will imply that G is metrisable by [20, Theorem 8.3]. Indeed, it follows from Tannaka–Krein duality [22] that the topology on G is the initial topology induced by its irreducible continuous unitary representations, and as such has a base given by sets of the form U(π1 , . . . , π n ; ε; t) := {s ∈ G : ‖π i (t) − π i (s)‖ < ε, i = 1, . . . , n},

where ε > 0, t ∈ G, and (π1 , H1 ), . . . , (π n , H n ) ∈ ̂ G. Hence, if ̂ G is countable, for every t ∈ G the sets U(π1 , . . . , π n ; 1/m; t) (m ∈ N , π1 , . . . , π n ∈ ̂ G) form a countable neighbourhood base at t, and so G is first-countable. Now suppose instead that G is metrisable. Then C(G) is separable. Since the infinity norm dominates the L 2 -norm for a compact space, and since C(G) is dense in L 2 (G), it follows that L 2 (G) is separable. By [21, Theorem 27.40] L 2 (G) ≅ ⨁ H π

⊕dimH π

,

π∈̂ G

which is clearly separable only if ̂ G is countable. Hence (c) implies (b). Next we show that (b) implies (a). Suppose that ̂ G is countable. By Theorem 3.3(ii) T(G) is a dense ideal in L 1 (G) so that, by Lemma 2.1, I ∩ T(G) = I for every closed left ideal I in L 1 (G). Fix a closed left ideal I in L 1 (G). By Theorem 3.2 there exist linear subspaces E π ⊂ H π (π ∈ ̂ G) such that I = {f ∈ L 1 (G) : π(f)(E π ) = 0, π ∈ ̂ G} .

By Theorem 3.3(iii), for each π ∈ ̂ G we have T π (G) ≅ M d π (C), where d π is the dimension of H π , and since I ∩ T π (G) is a left ideal in T π (G) there must be an idempotent P π ∈ T π (G) such that I ∩ T π (G) = T π (G) ∗ P π . Set α π = ‖P π ‖−1 if P π ≠ 0, and set α π = 0 otherwise. Enumerate ̂ G = {π1 , π2 , . . .}, and define ∞

g=∑

i=1

1 α π P π ∈ L 1 (G), i2 i i

which belongs to I because each P π i does, and I is closed. We claim that I = L 1 (G) ∗ g. Indeed, I ⊃ L 1 (G) ∗ g because g ∈ I. For the reverse inclusion we show that, for j ∈ N and ξ ∈ H π j , we have π j (f)(ξ) = 0 for

236 | White all f ∈ L 1 (G) ∗ g if and only if ξ ∈ E π j . The claim then follows from Theorem 3.2.

Indeed, if f ∈ L 1 (G) ∗ g then π j (f)(ξ) = 0 because f ∈ I. On the other hand if ξ ∈ H π j \ E π j then π j (P π j )(ξ) ≠ 0, whereas π i (P π j ) = 0 for i ≠ j by Theorem 3.3(i), which implies that π j (g)ξ = j12 α π j π j (P π j )(ξ) ≠ 0. This establishes the claim. Finally we show that (a) implies (b). Assume that L 1 (G) is topologically left Noetherian. Then there exist r ∈ N and g1 , . . . , g r ∈ L 1 (G) such that L 1 (G) = L 1 (G) ∗ g1 + ⋅ ⋅ ⋅ + L 1 (G) ∗ g r . (i)

For each n ∈ N there exist t n ∈ T(G) (i = 1, . . . , r) such that (i)

‖t n − g i ‖
0. Then, by the definition of Ξ, S1 , . . . , S m ∈ I, and y1 , . . . , y m ∈ E such that ‖x − (S1 y1 + ⋅ ⋅ ⋅ + S m y m )‖ < ε. Since T1 A(E) + ⋅ ⋅ ⋅ + T n A(E) is dense in I, we may in fact suppose that S1 , . . . , S m ∈ T1 A(E) + ⋅ ⋅ ⋅ + T n A(E), so that S1 y1 + ⋅ ⋅ ⋅ + S m y m ∈ im T1 + ⋅ ⋅ ⋅ + im T n . As ε was arbitrary we see that x ∈ im T1 + ⋅ ⋅ ⋅ + im T n . The result now follows.

242 | White We omit the proof of the following well known result. In any case, it can be proved in a similar fashion to Lemma 4.4. Lemma 4.8. Let E be a Banach space, and let F be any separable Banach space. Then there exists an approximable linear map from E to F with dense range. We can now prove the theorem. Theorem 4.9. Let E be a Banach space such that A(E) has a right approximate identity. Then the following are equivalent: (a) the Banach algebra A(E) is topologically right Noetherian; (b) every closed right ideal of A(E) is topologically principal; (c) the space E is separable. Proof. It is trivial that (b) implies (a). We show that (a) implies (c). Suppose that A(E) is topologically right Noetherian. Then A(E) = R(E) is topologically finitelygenerated so that, by Lemma 4.7, there exist n ∈ N and T1 , . . . , T n ∈ A(E) such that E = im T1 + ⋅ ⋅ ⋅ + im T n . Since each operator T i (i = 1, . . . , n) is compact, its image is separable, and hence so is E. Now suppose instead that E is separable, and let I be a closed right ideal of A(E). Then, by Theorem 4.6, I = R(F) for some F ∈ SUB(E). By Lemma 4.8 there ̂ (TA(E)) = im T = F, so exists T ∈ A(E) with im T = F. By Lemma 4.7 we have Ξ that, by Theorem 4.6, I = R(F) = TA(E). Since I was arbitrary, this shows that (c) implies (b).

We can now prove Theorem 1.5 as a special case of our results above. Proof of Theorem 1.5. Of course, under either hypothesis K(E) = A(E). By [10, Theorem 2.5 (ii)], K(E) has a left approximate identity whenever E has the approximation property. Similarly, by [17, Theorem 3.3] K(E) has a (bounded) twosided approximate identity whenever E󸀠 has the bounded approximation property. Hence the results follow from Theorem 4.5 and Theorem 4.9. Remark. Consider K(ℓ 1 ). Of course, (ℓ 1 )󸀠 ≅ ℓ ∞ , which has BAP by [37, Example 5(a), Chapter II E]. Hence, by Theorem 1.5, K(ℓ 1 ) is an example of a Banach algebra which is topologically right Noetherian, but not topologically left Noetherian.

We observe that, although it talks about algebraically finitely-generated left ideals, the argument given in [11, Corollary 3.2] actually proves that any Banach space E satisfying its hypothesis has the property that B(E) is not topologically left Noetherian. Indeed, the argument there is to demonstrate that there are more (maximal) closed left ideals in B(E) than there are finite n-tuples of operators.

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Hence not every closed left ideal can be topologically finitely-generated. This covers a large class of Banach spaces, including, for example, c0 , ℓ p for 1 ≤ p < ∞, L p [0, 1] for 1 < p < ∞, and many other spaces discussed in [11]. In the case that E is a reflexive Banach space, B(E) is a dual Banach algebra with predual ̂ E󸀠 . We shall show in the next section that, for a reflexive Banach space given by E⊗ E with the approximation property, B(E) is weak*-topologically left Noetherian whenever K(E) is. Hence in particular, many of the above examples which fail to be ‖ ⋅ ‖-topologically left Noetherian are weak*-topologically left Noetherian. We note however that it is possible for B(E) to be topologically left Noetherian, for an infinite-dimensional Banach space E. Let E AH be the Banach space with the property that B(E AH ) = C idE AH + K(E AH ) constructed by Argyros and Haydon in [3]. Since E AH is a predual of ℓ 1 , which has BAP, it satisfies the hypotheses of part (i) and (ii) of Theorem 1.5. Since B(E AH ) = K(E AH )♯ , Theorem 4.5 and Lemma 2.2(iii) imply that B(E AH ) is ‖ ⋅ ‖-topologically left and right Noetherian. Further examples come from [28], where the authors construct, for each countably infinite, compact metric space X, a predual of ℓ 1 , say E X , such that B(E X )/K(E X ) ≅ C(X). Since C(X) is topologically Noetherian for any compact metric space X, a similar argument shows that the Banach algebras B(E X ) are topologically left and right Noetherian. In all of these examples the Banach space is hereditarily indecomposable. There is no hereditarily indecomposable Banach space E for which we know that B(E) is not topologically left/right Noetherian.

5 Multiplier Algebras and Dual Banach Algebras In this section we consider those Banach algebras A whose multiplier algebra is a dual Banach algebra. We shall focus on the case in which A has a bounded approximate identity. Examples of such Banach algebras include L1 (G) for G a locally compact group, and B(E) for E a reflexive Banach space with AP. Other examples include the Figà-Talamanca–Herz algebras A p (G) for G a locally compact amenable group, and p ∈ (1, ∞), with the predual of M(A p (G)) given by PF p (G), the algebra of p-pseudo-functions of G. Also L 1 (G), where G is a locally compact quantum group in the sense of Kustermans and Vaes, fits into this setting whenever it has a bounded approximate identity [9]. We prove in Proposition 5.3 that, for a Banach algebra A satisfying a fairly mild condition, the multiplier algebra M(A) is weak*-topologically left Noetherian whenever A is ‖ ⋅ ‖-topologically left Noetherian. Corollary 1.6 then follows. We go on to prove that for a certain, more restrictive class of Banach algebras there is a

244 | White bijective correspondence between the closed left ideals of A and the weak*-closed left ideals of M(A) (Theorem 5.10). For background on multiplier algebras see one of [8, 9, 29]. We say that A is faithful if Ax = {0} implies x = 0 (x ∈ A), and also xA = {0} implies x = 0 (x ∈ A). We recall that the canonical map A → M(A) is injective if and only if A is faithful. When A has a bounded approximate identity, this map is bounded below, so that A is isomorphic to its image inside M(A). When this is the case we shall identify A with its image inside M(A). In this section, when we consider a linear functional applied to a vector, we shall often use a subscript to indicate the exact dual pairing. So for example, if A is a Banach algebra, and a ∈ A and f ∈ A󸀠 , we might write ⟨a, f⟩(A, A󸀠 ) or ⟨f, a⟩(A󸀠 , A) for the value of f applied to a. In [9] Daws considers Banach algebras whose multiplier algebras are also dual Banach algebras. We shall use the following consequence of Daws’ work, which essentially says that when such a Banach algebra has a bounded approximate identity, the multiplier and dual structures are compatible in a natural way. Theorem 5.1. Let A be a Banach algebra with a bounded approximate identity, and suppose that M(A) is a dual Banach algebra, with predual X. Then X may be identified with a closed A-submodule of A ⋅ A󸀠 ⋅ A in such a way that ⟨f ⋅ a, μ⟩(X, M(A)) = ⟨f, aμ⟩(A󸀠 , A) , ⟨a ⋅ f, μ⟩(X, M(A)) = ⟨f, μa⟩(A󸀠 , A) ,

(1) (2)

for all μ ∈ M(A), and all a ∈ A and f ∈ A󸀠 with f ⋅ a ∈ X/ a ⋅ f ∈ X respectively. We also have ⟨x, a⟩(X, M(A)) = ⟨x, a⟩(A󸀠 , A) , (3) for all x ∈ X and a ∈ A.

Proof. The fact that X may be identified with a closed A-submodule of A ⋅ A󸀠 ⋅ A follows immediately from [9, Theorem 7.9] and the remarks following it, and Equations (1) and (2) then follow by chasing through the definition of the map θ0 of that theorem. Equation (3) then follows from (1): given a ∈ A and x ∈ X, let b ∈ A and f ∈ A󸀠 satisfy x = f ⋅ b. Then ⟨f ⋅ b, a⟩(X, M(A)) = ⟨f, ba⟩(A󸀠 , A) = ⟨f ⋅ b, a⟩(A󸀠 , A) , as required. Remark. Commutative Banach algebras whose multiplier algebras are dual Banach algebras satisfying (1)/(2) were considered by Ülger in [35]. Since Ülger always as-

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245

sumes the existence of a bounded approximate identity, Theorem 5.1 allows the hypothesis of [35, Theorem 3.7] to be simplified slightly. We note the following. Lemma 5.2. Let A be a Banach algebra with a bounded approximate identity, such that M(A) is a dual Banach algebra. Then A is weak*-dense in M(A). Proof. Let X be the predual of M(A). Suppose x ∈ A⊥ ⊂ X. Then, by Theorem 5.1, we may identify X with a closed subspace of A󸀠 , and for all a ∈ A we have w∗ 0 = ⟨x, a⟩(X, M(A)) = ⟨x, a⟩(A󸀠 , A) , which implies that x = 0. Hence A = (A⊥ )⊥ = {0}⊥ = M(A). We now prove a result about weak*-topological left Noetherianity in this setting. Note that, by the previous lemma, the hypothesis is satisfied by any Banach algebra with a bounded approximate identity whose multiplier algebra is a dual Banach algebra. Proposition 5.3. Let A be a Banach algebra such that M(A) admits the structure of a dual Banach algebra in such a way that A is weak*-dense in M(A). Suppose that for every closed left ideal I in A there exists n ∈ N and there exist μ1 , . . . , μ n ∈ M(A) such that I = A♯ μ1 + ⋅ ⋅ ⋅ + A♯ μ n . Then M(A) is weak*-topologically left Noetherian. In particular, M(A) is weak*-topologically left Noetherian whenever A is ‖ ⋅ ‖-topologically left Noetherian. Proof. Let I be a weak*-closed left ideal of M(A). Since A is weak*-dense in M(A), which is unital, Lemma 2.1 implies that A∩I is weak*-dense in I. On the other hand, A ∩ I is a closed left ideal in A, so there exists n ∈ N, and there exist μ1 , . . . , μ n ∈ M(A) such that A ∩ I = A♯ μ1 + ⋅ ⋅ ⋅ + A♯ μ n . It follows that I = A♯ μ1 + ⋅ ⋅ ⋅ + A♯ μ n

w∗

= M(A)μ1 + ⋅ ⋅ ⋅ + M(A)μ n

w∗

.

As I was arbitrary the result follows. We are now able to prove Corollary 1.6 concerning the weak*-topological left/right Noetherianity for algebras of the form M(G) and B(E). Note that, by Proposition 3.5 and the discussion at the end of Section 4, these algebras are often not ‖ ⋅ ‖topologically left/right Noetherian Proof of Corollary 1.6. This follows from Proposition 5.3, Theorem 1.4, and Theorem 1.5. Definition 5.4. Let A be a Banach algebra. We say that A is a compliant Banach algebra if A is faithful and M(A) is a dual Banach algebra in such a way that, for

246 | White each a ∈ A, the maps M(A) → A given by μ 󳨃→ μa and μ 󳨃→ aμ are weak*-weakly continuous. In this article we shall consider the ideal structure of compliant Banach algebras, but we note that they appear to have interesting properties more broadly and are worthy of further study. In the papers [18] and [19] Hayati and Amini consider Connes amenability of certain multiplier algebras which are also dual Banach algebras. In our terminology, [19, Theorem 3.3] says that if A is a compliant Banach algebra with a bounded approximate identity, then A is amenable if and only if M(A) is Connes amenable. We have the following family of examples of compliant Banach algebras. Lemma 5.5. Let A be a Banach algebra with a bounded approximate identity which is Arens regular and an ideal in its bidual. Then A is a compliant Banach algebra. Proof. By [24, Theorem 3.9] A󸀠󸀠 as an algebra with Arens multiplication may be identified with M(A). Arens regularity implies that A󸀠󸀠 is a dual Banach algebra with predual A󸀠 . Given a ∈ A, the maps μ 󳨃→ μa and μ 󳨃→ aμ are weak*continuous as maps from A󸀠󸀠 to itself, and hence they are weak*-weakly continuous when considered as maps from A󸀠󸀠 to A. It follows from Lemma 5.5 that c0 (N) is an example of a compliant Banach algebra. A family of examples that will be important to us is the following: Corollary 5.6. Let E be a reflexive Banach space with the approximation property. Then K(E) is a compliant Banach algebra. Proof. By [38, Theorem 3] K(E) is Arens regular. Moreover K(E)󸀠󸀠 = B(E), so that we see that K(E) is an ideal in its bidual. Hence the result follows from the previous lemma. The following lemma is also useful for finding examples. Lemma 5.7. Let A be as in Theorem 5.1, and suppose that the identification of that theorem yields X = A ⋅ A󸀠 = A󸀠 ⋅ A. Then A is a compliant Banach algebra.

Proof. Let (μ α ) be a net in M(A) which converges to some μ ∈ M(A) in the weak*topology. Fix a ∈ A, and let f ∈ A󸀠 be arbitrary. Then lim⟨f, aμ α ⟩(A󸀠 , A) = lim⟨f ⋅ a, μ α ⟩(X, M(A)) = ⟨f ⋅ a, μ⟩(X, M(A)) = ⟨f, aμ⟩(A󸀠 , A) . α

α

This shows that the map μ 󳨃→ aμ is weak*-weakly continuous, and by an analogous argument so is the map μ 󳨃→ μa. As a was arbitrary this proves the lemma. Compliance is a fairly restrictive condition, as the next Proposition illustrates.

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247

Proposition 5.8. (i) Let A be a compliant Banach algebra, and let ◻ denote the first Arens product on A󸀠󸀠 . Then A is an ideal in (A󸀠󸀠 , ◻). (ii) Let K be a locally compact space. The Banach algebra C0 (K) is compliant if and only if K is discrete. (iii) Let G be a locally compact group. The Banach algebra L 1 (G) is compliant if and only if G is compact. Proof. (i) If A is compliant then, for every a ∈ A, the maps given by L a : b 󳨃→ ab

R a : b 󳨃→ ba

and

󸀠󸀠 󸀠󸀠 󸀠󸀠 are weakly compact. Hence L󸀠󸀠 a (A ), R a (A ) ⊂ A by [26, Theorem 3.5.8]. Given 󸀠󸀠 󸀠󸀠 a ∈ A we have L a : Ψ 󳨃→ a◻Ψ (Ψ ∈ A ), and we see that A is a right ideal in A󸀠󸀠 . Similarly it is a left ideal. (ii) Whenever K is discrete C0 (K) is Arens regular with bidual given by ℓ ∞ (K). Hence the algebra is compliant by Lemma 5.7. Now suppose instead that C0 (K) is compliant. The dual space of C0 (K) may be identified with M(K), and, as such, for each x ∈ K, we may define an element ε x ∈ C0 (K)󸀠󸀠 by

ε x : μ 󳨃→ ∫ 1 dμ {x}

(μ ∈ M(K)).

By part (i) we know that, for any x ∈ K and any f ∈ C0 (K), we have ε x ◻f ∈ C0 (K). We then calculate that, for μ ∈ M(K), we have ⟨f ◻ε x , μ⟩ = ⟨ε x , μ ⋅ f⟩ = ∫ f dμ = μ({x})f(x) {x}

so that ε x ◻f is equal to f(x) at x, and 0 everywhere else. Therefore, given any x ∈ K, by choosing any f ∈ C0 (K) not vanishing at x, we see that the point mass at x is continuous. Hence K is discrete. (iii) If L 1 (G) is compliant, then, by part (i) and [15], G is compact. Suppose instead that G is compact. By examining the maps in [9, Theorem 7.9] we see that the identification in Theorem 5.1 is the usual inclusion C(G) → L∞ (G). Then C(G) = L 1 (G) ⋅ L ∞ (G) = L ∞ (G) ⋅ L 1 (G) by [14, Proposition 2.39(d)]. Hence, by Lemma 5.7, L 1 (G) is compliant. For compliant Banach algebras there is a bijective correspondence between the closed left ideals of A and the weak*-closed left ideals of M(A) as we describe below in Theorem 5.10. The next section will be devoted to applications of this result. Lemma 5.9. Let I be a closed left ideal of a compliant Banach algebra A, and let w∗ μ ∈ I ⊂ M(A). Then Aμ ⊂ I.

248 | White Proof. Let (μ α ) be a net in I converging to μ in the weak*-topology and let a ∈ A. For each index α we have aμ α ∈ I. Since A is compliant, the net aμ α converges w weakly to aμ in A. Hence aμ ∈ I = I. As a was arbitrary, the result follows. Theorem 5.10. Let A be a compliant Banach algebra with a bounded approximate identity. The map w∗ I 󳨃→ I , defines a bijective correspondence between closed left ideals in A and weak*-closed left ideals in M(A). The inverse is given by J 󳨃→ A ∩ J, for J a weak*-closed left ideal in M(A). w∗

Proof. First we take an arbitrary closed left ideal I in A and show that A ∩ I = I. w∗ w∗ Certainly I ⊂ A ∩ I . Let a ∈ A ∩ I . Then by Lemma 5.9 we have Aa ⊂ I. Since A has a bounded approximate identity, this implies that a ∈ I. As a was arbitrary, w∗ we must have I = A ∩ I . It remains to show that, given a weak*-closed left ideal J of M(A), we have w∗ A ∩ J = J, and this follows from Lemma 2.1 and Lemma 5.2. Finally we show that for compliant Banach algebras the converse of Proposition 5.3 holds, so that weak*-topological left Noetherianity of M(A) can be characterised in terms of a ‖ ⋅ ‖-topological condition on A. Proposition 5.11. Let A be a compliant Banach algebra with a bounded approximate identity. Then M(A) is weak*-topologically left Noetherian if and only if every closed left ideal I in A has the form I = Aμ1 + ⋅ ⋅ ⋅ + Aμ n , for some n ∈ N, and some μ1 , . . . , μ n ∈ M(A). Proof. The “if” direction follows from Proposition 5.3 and Lemma 5.2. Conversely, suppose that M(A) is weak*-topologically left Noetherian, and let I be a closed left ideal of A. Then there exist n ∈ N and μ1 , . . . , μ n ∈ M(A) such that I

w∗

= M(A)μ1 + ⋅ ⋅ ⋅ + M(A)μ n

w∗

= Aμ1 + ⋅ ⋅ ⋅ + Aμ n

w∗

,

where we have used Lemma 5.2 to get the second equality. Hence, by applying Theorem 5.10 twice, we obtain I=I

w∗

The result follows.

∩ A = Aμ1 + ⋅ ⋅ ⋅ + Aμ n

w∗

∩ A = Aμ1 + ⋅ ⋅ ⋅ + Aμ n .

Left Ideals of Banach Algebras | 249

6 Some Classification Results In this section we use Theorem 5.10 to give classifications of the weak*-closed left ideals of M(G), for G a compact group, and of the weak*-closed left ideals of B(E), for E a reflexive Banach space with the approximation property. We then observe that this gives us some classification results for the closed right submodules of the preduals. Let G be a compact group and suppose the for each π ∈ ̂ G we have chosen a linear subspace E π ≤ H π . Then we define J[(E π )π∈̂G ] := {μ ∈ M(G) : π(μ)(E π ) = 0, π ∈ ̂ G} .

We shall show that these are exactly the weak*-closed left ideals of M(G). Lemma 6.1. Let G be a compact group and let E π ≤ H π (π ∈ ̂ G). Then span {ξ ∗π η : π ∈ ̂ G, ξ ∈ E π , η ∈ H π } = J[(E π )π∈̂G ]. ⊥

(1)

Proof. Routine calculation.

Theorem 6.2. Let G be a compact group. Then the weak*-closed left ideals of M(G) are given by J[(E π )π∈̂G ], as (E π )π∈̂G runs over the possible choices of linear subspaces E π ≤ H π (π ∈ ̂ G). Proof. By Lemma 6.1 each space J[(E π )π∈̂G ] is weak*-closed, and it is easily checked that it is a left ideal. Moreover, by Theorem 3.2 each closed left ideal of L 1 (G) has the form L 1 (G) ∩ J[(E π )π∈̂G ], for some choice of subspaces E π ≤ H π (π ∈ ̂ G). By Proposition 5.8, L 1 (G) is a compliant Banach algebra, so we may apply Proposition 5.10 to see that this must be the full set of weak*-closed left ideals.

Recall the definitions of RB(E) (F) and IB(E) (F) given respectively in Equations (2) and (3) of Section 4. Theorem 6.3. Let E be a reflexive Banach space with the approximation property. Then the weak*-closed left ideals of B(E) are exactly given by IB(E) (F), as F runs through SUB(E). The weak*-closed right ideals are given by RB(E) (F), as F runs through SUB(E). Proof. For any closed linear subspace F ⊂ E the left ideal IB(E) (F) is weak*-closed since we have IB(E) (F) = {x ⊗ λ : x ∈ F, λ ∈ E󸀠 }⊥ ,

̂ E󸀠 . Similarly we have where x ⊗ λ denotes an element of the predual E⊗

250 | White

RB(E) (F) = {x ⊗ λ : x ∈ E, λ ∈ F ⊥ }⊥ ,

so that these right ideals are weak*-closed. Observe that for F ∈ SUB(E), we have LB(E) (F ⊥ ) = {T ∈ B(E) : im T 󸀠 ⊂ F ⊥ } = {T ∈ B(E) : im T 󸀠 ⊂ F ⊥ }

= {T ∈ B(E) : (im T 󸀠 )⊥ ⊃ F} = {T ∈ B(E) : ker T ⊃ F} = IB(E) (F),

(2)

so that, by Theorem 4.1, every closed left ideal of K(E) can clearly be written as IB(E) (F) ∩ K(E) for some F ∈ SUB(E). By Corollary 5.6 K(E) is compliant, so we may apply Proposition 5.10 to see that every weak*-closed left ideal has the required form. A similar argument applies to the weak*-closed right ideals. Finally we show how, using the following proposition, we can describe the closed ̂ E󸀠 , for E a reflexive Banach space with AP. We also left/right submodules of E⊗ obtain a description of the left-translation-invariant closed subspaces of C(G), for G a compact group (compare with [1, Theorem 2]). Proposition 6.4. Let (A, X) be a dual Banach algebra. Then there is a bijective correspondence between the closed right A-submodules of X and the weak*-closed left ideals of A given by Y 󳨃→ Y ⊥ , for Y a closed right A-submodule of X. Proof. It is quickly checked that Y ⊥ is a left ideal, whenever Y is a right submodule, and that I⊥ is a right submodule whenever I is a left ideal. Equation (1) of Section 2 now tells us that the given correspondence is bijective, with inverse given by I 󳨃→ I⊥ , for I a weak*-closed left ideal. In the following corollary, given a Banach space E, and a closed subspaces F ⊂ E ̂ E󸀠 with the closure of the algebraic tensor product F⊗E󸀠 and D ⊂ E󸀠 , we identify F ⊗ ̂ E󸀠 , and similarly for E⊗ ̂ D. inside E⊗

Corollary 6.5. Let E be a reflexive Banach space with the approximation property. ̂ E󸀠 are given by Then the closed right B(E)-submodules of E⊗ ̂ E󸀠 F⊗

(F ∈ SUB(E)).

̂D E⊗

(D ∈ SUB(E󸀠 )).

The closed left B(E)-submodules are given by

Left Ideals of Banach Algebras | 251

̂ E󸀠 is a closed right submodule. Proof. Given F ∈ SUB(E) it is easily seen that F ⊗ ̂ E󸀠 )⊥ is a weak*-closed left ideal by Proposition 6.4. It is easily Hence I := (F ⊗ checked that E󸀠 ∘ I = F ⊥ , and hence I = LB(E) (F ⊥ ) = IB(E) (F) by (2). Since the correspondence given in Proposition 6.4 is bijective, and since by Theorem 6.3 every weak*-closed ideal has the form IB(E) (F) for some F, it must be that every ̂ E󸀠 for some F ∈ SUB(E). closed right submodule has the form F ⊗ The result about closed left submodules is proved analogously. Corollary 6.6. Let G be a compact group, and let X ⊂ C(G) be a closed linear subspace, which is invariant under left translation. Then there exists a choice of linear subspaces E π ≤ H π (π ∈ ̂ G) such that X = span {ξ ∗π η : π ∈ ̂ G, ξ ∈ E π , η ∈ H π } .

Proof. In fact, by the weak*-density of the discrete measures in M(G), the closed right submodules of C(G) coincide with the closed linear subspaces invariant under left translation (compare with [36, Lemma 3.3]). By Proposition 6.4 X has the form I⊥ , for some weak*-closed left ideal I of M(G). It now follows from Theorem 6.2 and Lemma 6.1 that X has the given form. Acknowledgement: This work was supported by the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03). The article is based on part of the author’s PhD thesis, and as such he would like to thank his doctoral supervisors Garth Dales and Niels Laustsen, as well as his examiners Gordon Blower and Tom Körner, for their careful reading of earlier versions of this material and their helpful comments. We would also like to thank Yemon Choi for some helpful email exchanges. Finally, we would like to thank the anonymous referee for his/her many insightful comments, and in particular for conjecturing Proposition 5.8(ii).

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