Approximate Identities and Factorization in Banach Modules
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768 II II
II IIIIIIII
Robert S. Doran Josef Wichmann
Approximate Identities and Factorization in Banach Modules
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Springer-Verlag Berlin Heidelberg New York 197 9
Authors Robert S. Doran Department of Mathematics Texas Christian University Fort Worth, Texas 76129 USA Josef Wichmann Department of Mathematical Sciences University of Petroleum & Minerals Dhahran, Saudi Arabia
AMS Subject Classifications (1970): 22B10, 43A20, 46H05, 46H25, 46 L05, 46L20 ISBN 3-540-09725-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09725-2 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
TO PAUL J. COHEN AND EDWIN HEWITT
PREFACE In recent years Banach algebras with an approximate identity have received an increasing amount of attention.
Many results known for
C*-algebras or for algebras with an identity have been extended to algebras with an approximate identity. since it includes the group algebra G
This is an important extension
LI(G)
of a locally compact group
and many other naturally occuring algebras. Surprisingly little is known about the approximate identities
themselves.
In this monograph we have tried to collect all basic results
about them with the aim of stimulating further research in this direction. As the main tool in the study of Banach algebras with bounded approximate identity we present Cohen-Hewitt's factorization theorem for Banach modules and its many refinements.
Recognizing that factorization theory
is a subject of great importance in its own right, we have included the most recent and up-to-date results in this area that we had knowledge of. The level of exposition should be appropriate for those who are familiar with basic real and complex analysis, the elementary theory of commutative Banach algebras, and first results concerning C*-algebras (say, a nodding acquaintence with the first two chapters of Dixmier [70]). Granted these, the monograph contains complete proofs, although we should warn the reader that some of the arguments are a bit tedious and will require some diligence on his part. algebras are over the compl~x field.
We have assumed throughout that all The interested reader can determine,
by examining a given definition or theorem, if the complexes can be replaced by the reals.
Vl Examples and counterexamples are discussed whenever we knew of them. There is much further research to be done, and we have indicated in the last chapter a number of unsolved p
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