Amenable Banach algebras. A panorama

✍ Scribed by Runde, Volker

• Publisher: Springer Nature
• Year: 2020
• Tongue: English
• Leaves: 468
• Series: Springer Monographs in Mathematics
• Category: Library

✦ Synopsis

This volume provides readers with a detailed introduction to the amenability of Banach algebras and locally compact groups. By encompassing important foundational material, contemporary research, and recent advancements, this monograph offers a state-of-the-art reference. It will appeal to anyone interested in questions of amenability, including those familiar with the author's previous volume Lectures on Amenability. Cornerstone topics are covered first: namely, the theory of amenability, its historical context, and key properties of amenable groups. This introduction leads to the amenability of Banach algebras, which is the main focus of the book. Dual Banach algebras are given an in-depth exploration, as are Banach spaces, Banach homological algebra, and more. By covering amenability's many applications, the author offers a simultaneously expansive and detailed treatment. Additionally, there are numerous exercises and notes at the end of every chapter that further elaborate on the chapter's contents. Because it covers both the basics and cutting edge research, Amenable Banach Algebras will be indispensable to both graduate students and researchers working in functional analysis, harmonic analysis, topological groups, and Banach algebras. Instructors seeking to design an advanced course around this subject will appreciate the student-friendly elements; a prerequisite of functional analysis, abstract harmonic analysis, and Banach algebra theory is assumed.;Preface -- List of Symbols -- Introduction -- 0 Paradoxical Decompositions -- 1 Amenable, Locally Compact Groups -- 2 Amenable Banach Algebras -- 3 Examples -- 4 Amenability-Like Properties -- 5 Dual Banach Algebras -- 6 Banach Homological Algebra -- 7 Operator Algebras on Hilbert Spaces -- 8 Operator Amenability -- A Banach Spaces -- B Banach Algebras -- C C*- and Neumann Algebras -- D Abstract Harmonic Analysis -- E Operator Spaces -- F Fourier and Fourier-Stieltjes Algebras -- References -- Index of Names -- Index.

✦ Table of Contents

Preface......Page 7
Introduction......Page 10
Contents......Page 13
0 Paradoxical Decompositions......Page 16
0.1 The Banach–Tarski Paradox......Page 17
0.2 Tarski's Theorem......Page 24
1.1 Invariant Means and Asymptotic Invariance Properties......Page 35
1.2 Hereditary Properties......Page 46
1.3 Uniformly Bounded Representations......Page 54
1.4 Leptin's Theorem......Page 59
1.5 Fixed Point Theorems......Page 62
2 Amenable Banach Algebras......Page 70
2.1 Derivations from Group Algebras......Page 71
2.2 Virtual and Approximate Diagonals......Page 79
2.3 Hereditary and Splitting Properties......Page 86
2.4 A First Look at Hochschild Cohomology......Page 100
3.1 Measure Algebras......Page 110
3.2 Fourier and Fourier–Stieltjes Algebras......Page 118
3.3 Algebras of Approximable Operators......Page 125
3.4 (Non-)Amenability of mathcalB(E)......Page 137
3.5 An Amenable Radical Banach Algebra......Page 147
4 Amenability-Like Properties......Page 163
4.1 Contractibility......Page 164
4.2 Weak Amenability......Page 167
4.3 Character Amenability......Page 174
4.4 Pseudo- and Approximate Amenability......Page 180
4.5 Biflatness and Biprojectivity......Page 192
5 Dual Banach Algebras......Page 212
5.1 Connes-Amenability for Dual Banach Algebras......Page 213
5.2 The Case of the Measure Algebra......Page 229
5.3 Connes-Amenability without a Normal, Virtual Diagonal......Page 241
5.4 Daws' Representation Theorem......Page 251
5.5 Connes-Amenability and Connes-Injectivity......Page 256
6.1 Projectivity......Page 269
6.2 Resolutions and Ext-Groups......Page 274
6.3 Flatness and Injectivity......Page 286
7 Operator Algebras on Hilbert Spaces......Page 294
7.1 Amenable von Neumann Algebras......Page 295
7.2 Injective von Neumann Algebras......Page 302
7.3 Nuclear Cast-Algebras......Page 307
7.4 Semidiscrete von Neumann Algebras......Page 310
7.5 Normal, Virtual Diagonals......Page 325
7.6 Commutative Operator Algebras......Page 337
7.7 An Amenable Operator Algebra Not Similar to a Cast-Algebra......Page 341
8.1 Operator Amenable, Completely Contractive Banach Algebras......Page 352
8.2 Fourier Algebras......Page 356
8.3 Fourier–Stieltjes Algebras......Page 363
A.1 Bases in Banach Spaces......Page 373
A.2 Approximation Properties......Page 374
A.3 The Radon–Nikodým Property......Page 375
A.4 Local Theory......Page 376
B.1 Spectra and Gelfand Theory......Page 380
B.2 Banach Modules and Bounded Approximate Identities......Page 382
B.3 Multiplier Algebras......Page 384
B.4 Prime and Primitive ideals......Page 385
B.5 Structure of Semiprime and Semisimple Banach Algebras......Page 388
C.1 ast-Algebras and -Homomorphisms......Page 390
C.2 Cast-Algebras......Page 391
C.3 Positivity in Cast-Algebras and Their Duals......Page 392
C.4 ast-Representations of Cast-Algebras......Page 394
C.5 von Neumann Algebras and Wast-Algebras......Page 396
C.6 Multipliers of Cast-Algebras......Page 398
C.7 Projections in von Neumann Algebras......Page 400
C.8 Tensor Products of Cast- and von Neumann Algebras......Page 402
C.9 Weights on von Neumann algebras......Page 403
D.1 Semitopological Semigroups and Locally Compact Groups......Page 406
D.2 The Group Algebra......Page 407
D.3 The Measure Algebra M(G)......Page 409
D.4 Other Banach Algebras Associated with Locally Compact Groups......Page 411
E.1 Concrete and Abstract Operator Spaces......Page 414
E.2 Completely Bounded Maps......Page 416
E.3 Duality......Page 418
E.4 The Projective Tensor Product of Operator Spaces......Page 420
E.5 Completely Contractive Banach Algebras......Page 421
F.1 Representations of Locally Compact Groups......Page 424
F.2 The Fourier Algebra......Page 425
F.3 The Fourier–Stieltjes Algebra......Page 428
F.4 Cosets, Idempotents, and Piecewise Affine Maps......Page 430
References......Page 432
Index of Symbols......Page 453
Index......Page 457
Index of Names......Page 467

✦ Subjects

Functional analysis;Harmonic analysis;Operator theory