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English Pages [114] Year 1974
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