Smooth Homogeneous Structures in Operator Theory

Daniel Beltita158488617X, 9781584886174, 9781420034806

Those who work in operator theory and the theory of operator algebras know how important geometric ideas and techniques are to success. Here Beltita (mathematics, the Romanian Academy) gives readers the background they need to understand this cycle, and also describes the newest research. He covers topological Lie algebras, Lie groups and their Lie algebras, enlargeability, smooth homogeneous spaces, quasimultiplicative maps, complex structures in homogeneous spaces, equivariant monotone operators, L*-ideals and equivariant monotone operators, homogeneous spaces of pseudo-restricted groups

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Table of contents :
SMOOTH HOMOGENEOUS STRUCTURES IN OPERATOR THEORY……Page 1
Contents……Page 3
Preface……Page 5
Introduction……Page 7
Motivation……Page 8
Some particular tools……Page 9
More detailed description of contents……Page 10
Lie theory……Page 14
1.1 Fundamentals……Page 15
Topological Lie algebras of vector fields……Page 17
1.2 Universal enveloping algebras……Page 18
1.3 The Baker-Campbell-Hausdor series……Page 21
1.4 Convergence of the Baker-Campbell-Hausdor series……Page 31
Notes……Page 36
2.1 Definition of Lie groups……Page 37
2.2 The Lie algebra of a Lie group……Page 42
Examples of Lie groups out of topological algebras……Page 47
2.3 Logarithmic derivatives……Page 49
2.4 The exponential map……Page 51
Example of non-regular Lie group……Page 53
2.5 Special features of Banach-Lie groups……Page 55
Notes……Page 66
3.1 Integrating Lie algebra homomorphisms……Page 67
3.2 Topological properties of certain Lie groups……Page 76
3.3 Enlargible Lie algebras……Page 86
Example of non-enlargible Banach-Lie algebra……Page 87
Notes……Page 88
Homogeneous spaces……Page 89
4.1 Basic facts on smooth homogeneous spaces……Page 90
Banach-Lie subgroups……Page 91
Algebraic subgroups……Page 95
Smooth quotient spaces……Page 99
4.2 Symplectic homogeneous spaces……Page 103
Unitary orbits of self-adjoint operators……Page 109
Unitary orbits of spectral measures……Page 111
Unitary orbits of group representations……Page 112
Notes……Page 113
5.1 Supports, convolution, and quasimultiplicativity……Page 115
5.2 Separate parts of supports……Page 120
Some intertwining properties of quasimultiplicative maps……Page 124
5.3 Hermitian maps……Page 131
Notes……Page 135
6.1 General results……Page 136
6.2 Pseudo-Kahler manifolds……Page 141
6.3 Flag manifolds in Banach algebras……Page 146
Notes……Page 152
Equivariant monotone operators and Kahler structures……Page 153
7.1 Definition of equivariant monotone operators……Page 154
7.2 H*-algebras and L*-algebras……Page 158
7.3 Equivariant monotone operators as reproducing kernels……Page 164
7.4 H*-ideals of H*-algebras……Page 170
7.5 Elementary properties of H*-ideals……Page 174
Notes……Page 176
8.1 From ideals to operators……Page 178
8.2 From operators to ideals……Page 181
8.3 Parameterizing L*-ideals……Page 190
Elliptic Banach-Lie algebras……Page 194
8.4 Representations of automorphism groups……Page 196
8.5 Applications to enlargibility……Page 198
Notes……Page 200
9.1 Pseudo-restricted algebras and groups……Page 202
Cocycles of pseudo-restricted algebras……Page 209
9.2 Complex polarizations……Page 211
9.3 Kahler polarizations……Page 214
9.4 Admissible pairs of operator ideals……Page 220
9.5 Some Kahler homogeneous spaces……Page 225
Notes……Page 231
Appendices……Page 232
A.1 Locally convex spaces and algebras……Page 233
Complexifications and continuous inverse algebras……Page 235
A.2 Differential calculus and in nite-dimensional holomorphy……Page 240
Analytic mappings……Page 244
A.3 Smooth manifolds……Page 248
Vector fields……Page 252
Differential forms……Page 257
Notes……Page 258
Appendix B: Basic Differential Equations of Lie Theory……Page 260
Notes……Page 274
Appendix C: Topological Groups……Page 276
Notes……Page 285
References……Page 286