A course in commutative Banach algebras / Eberhard Kaniuth.

Format:

Book

Author/Creator:

Kaniuth, Eberhard.

Series:

Graduate texts in mathematics ; 246.

Graduate texts in mathematics ; 246

Language:

English

Subjects (All):

Banach algebras.

Commutative algebra.

Physical Description:

xii, 353 pages : illustrations ; 25 cm.

Place of Publication:

New York ; London : Springer, 2009.

Summary:

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 is formed by 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 course in harmonic analysis based on Gelfand theory.

Contents:

1 General Theory of Banach Algebras 1

1.1 Basic definitions and examples 1

1.2 The spectrum of a Banach algebra element 8

1.3 L[superscript 1]-algebras and Beurling algebras 18

1.4 Ideals and multiplier algebras 22

1.5 Tensor products of Banach algebras 30

2 Gelfand Theory 43

2.1 Multiplicative linear functionals 44

2.2 The Gelfand representation 52

2.3 Finitely generated commutative Banach algebras 60

2.4 Commutative C*-algebras 66

2.5 The uniform algebras P(X) and R(X) 73

2.6 The structure space of A(X) 84

2.7 The Gelfand representation of L[superscript 1](G) 89

2.8 Beurling algebras L[superscript 1](G, [omega]) 99

2.9 The Fourier algebra of a locally compact group 107

2.10 The algebra of almost periodic functions 111

2.11 Structure spaces of tensor products 120

3 Functional Calculus, Shilov Boundary, and Applications 139

3.1 The holomorphic functional calculus 140

3.2 Some applications of the functional calculus 152

3.3 The Shilov boundary 158

3.4 Topological divisors of zero 169

3.5 Shilov's idempotent theorem and applications 178

4 Regularity and Related Properties 193

4.1 The hull-kernel topology 194

4.2 Regular commutative Banach algebras 198

4.3 The greatest regular subalgebra 207

4.4 Regularity of L[superscript 1](G) 213

4.5 Spectral extension properties 222

4.6 The unique uniform norm property 229

4.7 Regularity of Beurling algebras 236

5 Spectral Synthesis and Ideal Theory 253

5.1 Basic notions and local membership 254

5.2 Spectral sets and Ditkin sets 260

5.3 Ideals in C[superscript n] [0, 1] 269

5.4 Spectral synthesis in the Mirkil algebra 278

5.5 Spectral sets and Ditkin sets for L[superscript 1](G) 288

5.6 Ideals with bounded approximate identities in L[superscript 1](G) 295

5.7 On spectral synthesis in projective tensor products 305

A.1 Topology 319

A.2 Functional analysis 321

A.3 Measure and integration 325

A.4 Haar measure and convolution on locally compact groups 327

A.5 The Pontryagin duality theorem 332

A.6 The coset ring of an Abelian group 336.

Notes:

Includes bibliographical references (pages [343]-348) and index.

ISBN:

9780387724751

0387724753

OCLC:

178312763