Graduate Texts in Mathematics

By: Eberhard Kaniuth

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Hardcover | 11 November 2008

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368 Pages

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23.5 x 15.88 x 1.91

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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.

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"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)

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You Can Find This Book In

Non-Fiction Mathematics Algebra Topology Calculus & Mathematical Analysis Functional Analysis & Transforms

General Theory of Banach Algebras	p. 1
Basic definitions and examples	p. 1
The spectrum of a Banach algebra element	p. 8
L[superscript 1]-algebras and Beurling algebras	p. 18
Ideals and multiplier algebras	p. 22
Tensor products of Banach algebras	p. 30
Exercises	p. 35
Notes and references	p. 41
Gelfand Theory	p. 43
Multiplicative linear functionals	p. 44
The Gelfand representation	p. 52
Finitely generated commutative Banach algebras	p. 60
Commutative C*-algebras	p. 66
The uniform algebras P(X) and R(X)	p. 73
The structure space of A(X)	p. 84
The Gelfand representation of L[superscript 1](G)	p. 89
Beurling algebras L[superscript 1](G, [omega])	p. 99
The Fourier algebra of a locally compact group	p. 107
The algebra of almost periodic functions	p. 111
Structure spaces of tensor products	p. 120
Exercises	p. 125
Notes and references	p. 135
Functional Calculus, Shilov Boundary, and Applications	p. 139
The holomorphic functional calculus	p. 140
Some applications of the functional calculus	p. 152
The Shilov boundary	p. 158
Topological divisors of zero	p. 169
Shilov's idempotent theorem and applications	p. 178
Exercises	p. 185
Notes and references	p. 190
Regularity and Related Properties	p. 193
The hull-kernel topology	p. 194
Regular commutative Banach algebras	p. 198
The greatest regular subalgebra	p. 207
Regularity of L[superscript 1](G)	p. 213
Spectral extension properties	p. 222
The unique uniform norm property	p. 229
Regularity of Beurling algebras	p. 236
Exercises	p. 245
Notes and references	p. 251
Spectral Synthesis and Ideal Theory	p. 253
Basic notions and local membership	p. 254
Spectral sets and Ditkin sets	p. 260
Ideals in C[superscript n] [0, 1]	p. 269
Spectral synthesis in the Mirkil algebra	p. 278
Spectral sets and Ditkin sets for L[superscript 1](G)	p. 288
Ideals with bounded approximate identities in L[superscript 1](G)	p. 295
On spectral synthesis in projective tensor products	p. 305
Exercises	p. 310
Notes and references	p. 316
Appendix	p. 319
Topology	p. 319
Functional analysis	p. 321
Measure and integration	p. 325
Haar measure and convolution on locally compact groups	p. 327
The Pontryagin duality theorem	p. 332
The coset ring of an Abelian group	p. 336
References	p. 343
Index	p. 349
Table of Contents provided by Ingram. All Rights Reserved.

Graduate Texts in Mathematics

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