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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 Gelfand's theory, regularity and spectral synthesis. Special emphasis is placed on applications in abstract harmonic analysis and on treating many special classes of commutative Banach algebras, such as uniform algebras, group algebras and Beurling algebras, and tensor products. Detailed proofs and a variety of exercises are given. The book aims at graduate students and can be used as a text for courses on Banach algebras, with various possible specializations, or a Gelfand theory based course in harmonic analysis.
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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)
"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)