Fourier and Fourier-Stieltjes Algebras on Locally Compact Groups

Cover......Page 1
Title page......Page 4
Contents......Page 8
Preface......Page 10
Acknowledgments......Page 12
1.1. Banach algebras and Gelfand theory of commutative Banach algebras......Page 14
1.2. Locally compact groups and examples......Page 19
1.3. Haar measure and group algebra......Page 25
1.4. Unitary representations and positive definite functions......Page 31
1.5. Abelian locally compact groups......Page 37
1.6. Representations and positive definite functionals......Page 41
1.7. Weak containment of representations......Page 43
1.8. Amenable locally compact groups......Page 46
Chapter 2. Basic Theory of Fourier and Fourier-Stieltjes Algebras......Page 50
2.1. The Fourier-Stieltjes algebra ()......Page 51
2.2. Functorial properties of ()......Page 59
2.3. The Fourier algebra (), its spectrum and its dual space......Page 63
2.4. Functorial properties and a description of ()......Page 70
2.5. The support of operators in ()......Page 73
2.6. The restriction map from () onto ()......Page 79
2.7. Existence of bounded approximate identities......Page 85
2.8. The subspaces _{}() of ()......Page 91
2.9. Some examples......Page 96
2.10. Notes and references......Page 99
3.1. Host’s idempotent theorem......Page 104
3.2. Isometric isomorphisms between Fourier-Stieltjes algebras......Page 109
3.3. Homomorphisms between Fourier and Fourier-Stieltjes algebras......Page 114
3.4. Invariant subalgebras of () and subgroups of ......Page 120
3.5. Invariant subalgebras of () and ()......Page 126
3.6. Comparison of (₁)̂⊗(₂) and (₁×₂)......Page 130
3.7. The *-topology and other topologies on ()......Page 134
3.8. Notes and references......Page 140
4.1. () as a completely contractive Banach algebra......Page 142
4.2. Operator amenability of ()......Page 145
4.3. Operator weak amenability of ()......Page 151
4.4. The flip map and the antidiagonal......Page 153
4.5. Amenability and weak amenability of () and of ¹()......Page 157
4.6. Notes and references......Page 165
5.1. Multipliers of ()......Page 166
5.2. (())=() implies amenability of : The discrete case......Page 173
5.3. (())=() implies amenability of : The nondiscrete case......Page 180
5.4. Completely bounded multipliers......Page 192
5.5. Uniformly bounded representations and multipliers......Page 199
5.6. Multiplier bounded approximate identities in ()......Page 204
5.7. Examples: Free groups and (2,ℝ)......Page 208
5.8. Notes and references......Page 215
Chapter 6. Spectral Synthesis and Ideal Theory......Page 218
6.1. Sets of synthesis and Ditkin sets......Page 219
6.2. Malliavin’s theorem for ()......Page 223
6.3. Injection theorems for spectral sets and Ditkin sets......Page 224
6.4. A projection theorem for local spectral sets......Page 227
6.5. Bounded approximate identities I: Ideals......Page 233
6.6. Bounded approximate identities II......Page 241
6.7. Notes and references......Page 247
Chapter 7. Extension and Separation Properties of Positive Definite Functions......Page 250
7.1. The extension property: Basic facts......Page 251
7.2. Extending from normal subgroups......Page 255
7.3. Connected groups and SIN-groups......Page 259
7.4. Nilpotent groups and 2-step solvable examples......Page 263
7.5. The separation property: Basic facts and examples......Page 270
7.6. The separation property: Nilpotent Groups......Page 277
7.7. The separation property: Almost connected groups......Page 281
7.8. Notes and references......Page 286
A.1. The closed coset ring......Page 290
A.2. Amenability and weak amenability of Banach algebras......Page 293
A.3. Operator spaces......Page 295
A.4. Operator amenability......Page 297
A.5. Operator weak amenability......Page 300
Bibliography......Page 304
Index......Page 316
Back Cover......Page 321