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ψ Banach modules over Banach algebras

A.W. Μ. Graven

BANACH MODULES OVER BANACH ALGEBRAS

Promotor:

Prof.Dr. A.C.M, van Rooij

BANACH MODULES OVER BANACH ALGEBRAS

PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE WISKUNDE EN NATUURWETENSCHAPPEN AAN DE KATHOLIEKE UNIVERSITEIT TE NIJMEGEN, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF.Mr. F.J.F.M.DUYNSTEE, VOLGENS BESLUIT VAN HET COLLEGE VAN DECANEN IN HET OPENBAAR TE VERDEDIGEN OP VRIJDAG 8 NOVEMBER 1974, DES MIDDAGS TE 2 UUR PRECIES.

DOOR ALPHONS WILLEM MICHIEL GRAVEN GEBOREN TE GELEEN

1974 KRIPS REPRO-MEPPEL

I wish to express my appreciation to my wife, Trees, for her excellent typing of the thesis.

Aan mijn ouders Voor Trees en Jerome

CONTENTS

Introduction and summary

9

Chapter 1: Preliminaries 1.1: Notation

12

1.2: Definitions and examples

14

1.3: Tensor products of modules

19

1.4: Tensor products of multipliers

28

Chapter 2: Modules over Banach algebras with approximate identities 2.1: General properties

34

2.2: A negative answer to Rieffel's Question

42

Chapter 3: G-modules and L.(G)-modules 3.1: G-modules

45

3.2: Tensor products and multipliers of L.(G)-modules

50

3.3: 1-dimensional submodules

66

Chapter 4: Injective and projective Banach modules 4.1: Free and projective Banach modules

76

4.2: Injective Banach modules

84

4.3: Injective and projective L.(G)-modules

94

References

102

Index of symbols

104

Index of terms

105

Samenvatting

106

Curriculum Vitae

107

INTRODUCTION AND SUMMARY

This thesis is concerned with Banach spaces which are modules over Banach algebras. If A is a Banach algebra, then by a left (right) Banach A-module we mean a Banach space, V, which is a left (right) A-module in the algebraic sense, and for which | |av| | for all x.yev, f e L (G). 47

(ii) if W is a submodule of V, then the projection of V onto W is an L.(G)-multiplier. Proof.

Remark that we may suppose that V is a left L (G)-module. If not

consider V. (i) By Cor. 2.1.14 V is the Hilbert space sum of V and V . Take ХіУб . Then χ = χ e + χ о and y' = y' e + y'o with χ e",ys € V e and χ ο'-Ό ,y € V о. " By Th. 3.1.6 there is a unique shift in V such that V is a module with shift. With this shift u •+ u is an isometry from V to V . J s e e Therefore = for a l l u , v € V , s E G . Hence = ' s s e " ds

=

/f(s)ds = = < x e , ; f 4 s ) ( y e ) s . 1 d s > = for a l l f £ L-CG). е е 1 (ii) Let W be a submodule of V and let Ρ be the projection from V onto W. Let ve V, f€ L CG) . P(f*v) = f*Pv if and only if = 0 for all w € W . However = f*g from L (G) to L (G) . Proof. The proof is obtained from Cor. 2.1.22, because L (G) is the dual space of an essential L (G)-module. Ζ. 2.2 Theorem.

β

For 1 · f*j

from L. (G) χ A(G) into A(G) . The map к -*• (к')* is an L. (G)-isomorphism from L (G) to A(G) . Proof.

A(G) = L (G) * L (G) implies f*keA(G) if f € L (G), keA(G).

Applying Fubini's theorem we find (f*k) (s) = /rf(t

)k(ts)dt =

;G;6f(t'1)k(Y)7(ti')dYdt = f&{fGf^-1)ñt)dt)k{y)V(J)dy

=

ƒ£(£·)A (Y)k(Y)7(i)dY = ((f^'kì'Cs), forali f e L ^ G ) , k Ê L ^ G ) , s C G . Therefore, if f € L^ (G) and к e L (б) then f*k = ( ( f T k ) * , and

||f*k|l A = H c f j - k l l j i I I C f i l L l l k l l i £ llf'lljllkl^. so l|f*k|lA 1 I Idilli U I L · The rest is straightforward. We turn now to the problem of representing

0

L (G)e L (G) and C(G)e L (G) 55

as function spaces. 3.2.12

Theorem.

Let G be a compact Abelian group. If 2