A Course in Commutative Banach Algebras
Requiring only a basic knowledge of functional analysis, topology, complex analysis, measure theory and group theory, this book provides a thorough and self-contained introduction to the theory of commutative Banach algebras. The core are chapters on Gelf
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246 Editorial Board
S. Axler K.A. Ribet
For other titles published in this series, go to www.springer.com/series/136
Eberhard Kaniuth
A Course in Commutative Banach Algebras
123
Eberhard Kaniuth Institute of Mathematics Paderborn University Paderborn, Germany [email protected]
Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
ISBN: 978-0-387-72475-1 DOI: 10.1007/978-0-387-72476-8
K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA [email protected]
e-ISBN: 978-0-387-72476-8
Library of Congress Control Number: 2008937945 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com
Dedicated to my wife Ursula
Preface
Banach algebras are Banach spaces equipped with a continuous multiplication. In rough terms, there are three types of them: algebras of bounded linear operators on Banach spaces with composition and the operator norm, algebras consisting of bounded continuous functions on topological spaces with pointwise product and the uniform norm, and algebras of integrable functions on locally compact groups with convolution as multiplication. These all play a key role in modern analysis. Much of operator theory is best approached from a Banach algebra point of view and many questions in complex analysis (such as approximation by polynomials or rational functions in specific domains) are best understood within the framework of Banach algebras. Also, the study of a locally compact Abelian group is closely related to the study of the group algebra L1 (G). There exist a rich literature and excellent texts on each single class of Banach algebras, notably on uniform algebras and on operator algebras. This work is intended as a textbook which provides a thorough introduction to the theory of commutative Banach algebras and stresses the applications to commutative harmonic analysis while also touching on uniform algebras. In this sense and purpose the book resembles Larsen’s classical text [75] which shares many themes and has been a valuable resource. However, for advanced graduate students and researchers I have covered several topics which have not been published in books before, including some journal articles. The reader is exp
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