Course in Commutative Banach Algebras

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.

From the reviews:“A course in commutative Banach algebras is the outgrowth of several graduate courses the author has taught. … The beginning of each chapter sets the stage for what is to follow and each concludes with notes and references. … This well-written book is a valuable resource for anyone working in the area of commutative Banach algebras. The book is very readable … . The text is well developed, the material is well motivated, and the book has an extensive list of references for further reading.” (Pamela Gorkin, Mathematical Reviews, Issue 2010 d)“This is a very well written book devoted to the theory of commutative Banach algebras over the complex field C, with emphasis on applications to harmonic analysis. … The book contains a large collection of problems at the end of each chapter, and a very useful appendix on point set topology, functional analysis, measure theory, Haar measure and the Pontryagin duality theorem. It will be a valuable resource for graduate students and researchers in Banach algebra theory and abstract harmonic analysis.” (Anthony To-Ming Lau, Zentralblatt MATH, Vol. 1190, 2010)“Kaniuth’s book … is a masterpiece of exposition. The theory is developed in five big chapters, each ending with a copious collection of interesting exercises and notes and references both to a comprehensive bibliography and an appendix reprising key background results … . This clearly and carefully written book … is an excellent addition to the literature. … a companion text both to a general course on Banach algebras and to a more advanced course specialising in their application to harmonic analysis.” (Nick Lord, The Mathematical Gazette, Vol. 94 (531), November, 2010)