Banach Algebras And The General Theory Of Algebras Volume 1 Algebras And Banach Algebras

This is the first volume of a two volume set that provides a modern account of basic Banach algebra theory including all known results on general Banach *-algebras. This account emphasizes the role of *-algebraic structure and explores the algebraic results that underlie the theory of Banach algebras and *-algebras. The first volume, which contains previously unpublished results, is an independent, self-contained reference on Banach algebra theory. Each topic is treated in the maximum interesting generality within the framework of some class of complex algebras rather than topological algebras. Proofs are presented in complete detail at a level accessible to graduate students. The book contains a wealth of historical comments, background material, examples, particularly in noncommutative harmonic analysis, and an extensive bibliography. Volume II is forthcoming.

This second of two volumes gives a modern exposition of the theory of Banach algebras.

This second volume of a two-volume set provides a modern account of basic Banach algebra theory including all known results on general Banach *-algebras. The author emphasizes the roles of *-algebra structure and explores the algebraic results that underlie the theory of Banach algebras and *-algebras. Proofs are presented in complete detail at a level accessible to graduate students. The books contain a wealth of historical comments, background material, examples, and an extensive bibliography. Together they constitute the standard reference for the general theory of *-algebras. This second volume deals with *-algebras. Noteworthy chapters develop the theory of *-algebras without additional restrictions, going well beyond what has been proved previously in this context and describe locally compact groups and the *-algebras related to them.

Table of contents

Banach algebras are Banach spaces equipped with a continuous multipli- tion. In roughterms,there arethree types ofthem:algebrasofboundedlinear operators on Banach spaces with composition and the operator norm, al- bras consisting of bounded continuous functions on topological spaces with pointwise product and the uniform norm, and algebrasof 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 speci?c - mains) are best understood within the framework of Banach algebras. Also, the study of a locally compact Abelian group is closely related to the study 1 of the group algebra L (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.