Introduction to Banach Algebras, Operators, and Harmonic Analysis

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Cambridge University Press, Nov 13, 2003 - Mathematics - 324 pages
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This work has arisen from lecture courses given by the authors on important topics within functional analysis. The authors, who are all leading researchers, give introductions to their subjects at a level ideal for beginning graduate students, and others interested in the subject. The collection has been carefully edited so as to form a coherent and accessible introduction to current research topics. The first chapter by Professor Dales introduces the general theory of Banach algebras, which serves as a background to the remaining material. Dr Willis then studies a centrally important Banach algebra, the group algebra of a locally compact group. The remaining chapters are devoted to Banach algebras of operators on Banach spaces: Professor Eschmeier gives all the background for the exciting topic of invariant subspaces of operators, and discusses some key open problems; Dr Laursen and Professor Aiena discuss local spectral theory for operators, leading into Fredholm theory.
 

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Contents

1 Definitions and examples
3
2 Ideals and the spectrum
12
3 Gelfand theory
20
4 The functional calculus
30
5 Automatic continuity of homomorphisms
38
6 Modules and derivations
48
7 Cohomology
58
Harmonic analysis and amenability George Awills
73
18 Invariant subspaces for subdecomposable operators
171
19 Reflexivity of operator algebras
178
Invariant subspaces for commuting contractions
186
Appendix to Part III
193
Local spectral theory Kjeld Bagger Lausen
199
21 Basic notions from operator theory
201
22 Classes of decomposable operators
212
Duality theory
226

8 Locally compact groups
75
9 Group algebras and representations
86
10 Convolution operators
98
11 Amenable groups
109
12 Harmonic analysis and automatic continuity
121
Invariant subspaces JŐrg Eschmeier
135
13 Compact operators
137
14 Unitary dilations and the Hfunctional calculus
143
15 Hyperinvariant subspaces
154
16 Invariant subspaces for contractions
160
17 Invariant subspaces for subnormal operators
166
24 Preservation of spectra and index
230
25Multipliers on commutative Banach algebras
241
Appendix to Part IV
254
Singlevalued extension property and Fredholm theory Pietro Aiena
265
26 Semiregular operators
267
27 The singlevalued extension property
285
28 SVEP for semiFredholm operators
298
Index of symbols
319
Subject index
321
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