| Description |
1 online resource (xxi, 778 pages) |
| Series |
De Gruyter expositions in mathematics, 0938-6572 ; 28 |
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De Gruyter expositions in mathematics ; 28. 0938-6572
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| Contents |
A. Abstract Representation Theory -- Chapter I. Reproducing Kernel Spaces 3 -- I.1. Operator-Valued Positive Definite Kernels 3 -- I.2. The Cone of Positive Definite Kernels 14 -- Chapter II. Representations of Involutive Semigroups 20 -- II. 1. Involutive Semigroups 21 -- II. 2. Bounded Representations 24 -- II. 3. Hermitian Representations 29 -- II. 4. Representations on Reproducing Kernel Spaces 34 -- Chapter III. Positive Definite Functions on Involutive Semigroups 52 -- III. 1. Positive Definite Functions -- the Discrete Case 53 -- III. 2. Enveloping C*-algebras 68 -- III. 3. Multiplicity Free Representations 80 -- Chapter IV. Continuous and Holomorphic Representations 99 -- IV. 1. Continuous Representations and Positive Definite Functions 99 -- IV. 2. Holomorphic Representations of Involutive Semigroups 119 -- B. Convex Geometry and Representations of Vector Spaces -- Chapter V. Convex Sets and Convex Functions 125 -- V.1. Convex Sets and Cones 126 -- V.2. Finite Reflection Groups and Convex Sets 138 -- V.3. Convex Functions and Fenchel Duality 147 -- V.4. Laplace Transforms 163 -- V.5. The Characteristic Function of a Convex Set 174 -- Chapter VI. Representations of Cones and Tubes 184 -- VI. 1. Commutative Representation Theory 185 -- VI. 2. Representations of Cones 195 -- VI. 3. Holomorphic Representations of Tubes 205 -- VI. 4. Realization of Cyclic Representations by Holomorphic Functions 209 -- VI. 5. Holomorphic Extensions of Unitary Representations 214 -- C. Convex Geometry of Lie Algebras -- Chapter VII. Convexity in Lie Algebras 221 -- VII. 1. Compactly Embedded Cartan Subalgebras 222 -- VII. 2. Root Decompositions 231 -- VII. 3. Lie Algebras With Many Invariant Convex Sets 251 -- Chapter VIII. Convexity Theorems and Their Applications 265 -- VIII. 1. Admissible Coadjoint Orbits and Convexity Theorems 266 -- VIII. 2. The Structure of Admissible Lie Algebras 292 -- VIII. 3. Invariant Elliptic Cones in Lie Algebras 306 -- D. Highest Weight Representations of Lie Algebras, Lie Groups, and Semigroups -- Chapter IX. Unitary Highest Weight Representations: Algebraic Theory 327 -- IX. 1. Generalized Highest Weight Representations 328 -- IX. 2. Positive Complex Polarizations 344 -- IX. 3. Highest Weight Modules of Finite-Dimensional Lie Algebras 356 -- IX. 4. The Metaplectic Factorization 361 -- IX. 5. Unitary Highest Weight Representations of Hermitian Lie Algebras 374 -- Chapter X. Unitary Highest Weight Representations: Analytic Theory 387 -- X.1. The Convex Moment Set of a Unitary Representation 388 -- X.2. Irreducible Unitary Representations 394 -- X.3. The Metaplectic Representation and Its Applications 400 -- X.4. Special Properties of Unitary Highest Weight Representations 411 -- X.5. Moment Sets for C*-algebras 419 -- X.6. Moment Sets for Group Representations 428 -- Chapter XI. Complex Ol'shanskii Semigroups and Their Representations 442 -- XI. 1. Lawson's Theorem on Ol'shanskii Semigroups 443 -- XI. 2. Holomorphic Extension of Unitary Representations 457 -- XI. 3. Holomorphic Representations of Ol'shanskii Semigroups 464 -- XI. 4. Irreducible Holomorphic Representations 470 -- XI. 5. Gelfand-Raikov Theorems for Ol'shanskii Semigroups 476 -- XI. 6. Decomposition and Characters of Holomorphic Representations 477 -- Chapter XII. Realization of Highest Weight Representations on Complex Domains 493 -- XII. 1. The Structure of Groups of Harish-Chandra Type 494 -- XII. 2. Representations of Groups of Harish-Chandra Type 514 -- XII. 3. The Compression Semigroup and Its Representations 524 -- XII. 5. Hilbert Spaces of Square Integrable Holomorphic Functions 538 -- E. Complex Geometry and Representation Theory -- Chapter XIII. Complex and Convex Geometry of Complex Semigroups 557 -- XIII. 1. Locally Convex Functions and Local Recession Cones 559 -- XIII. 2. Invariant Convex Sets and Functions in Lie Algebras 563 -- XIII. 3. Calculations in Low-Dimensional Cases 571 -- XIII. 4. Biinvariant Plurisubharmonic Functions 576 -- XIII. 5. Complex Semigroups and Stein Manifolds 586 -- XIII. 6. Biinvariant Domains of Holomorphy 595 -- Chapter XIV. Biinvariant Hilbert Spaces and Hardy Spaces on Complex Semigroups 600 -- XIV. 1. Biinvariant Hilbert Spaces 601 -- XIV. 2. Hardy Spaces Defined by Sup-Norms 608 -- XIV. 3. Hardy Spaces Defined by Square Integrability 616 -- XIV. 4. The Fine Structure of Hardy Spaces 623 -- Chapter XV. Coherent State Representations 645 -- XV. 1. Complex Structures on Homogeneous Spaces 646 -- XV. 2. Coherent State Representations 650 -- XV. 3. Heisenberg's Uncertainty Principle and Coherent States 656 -- Appendix I. Bounded Operators on Hilbert Spaces 665 -- Appendix II. Spectral Measures and Unbounded Operators 677 -- Appendix III. Holomorphic Functions on Infinite-Dimensional Spaces 686 -- Appendix IV. Symplectic Geometry 694 -- Appendix V. Simple Modules of p-Length 2 705 -- Appendix VI. Symplectic Modules of Convex Type 715 -- Appendix VII. Square Integrable Representations of Locally Compact Groups 727 -- Appendix VIII. The Stone-von Neumann-Mackey Theorem 742 |
| Bibliography |
Includes bibliographical references (pages 751-766) and index |
| Notes |
Print version record |
| Subject |
Lie groups.
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Representations of groups.
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Convex functions.
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MATHEMATICS -- Algebra -- Intermediate.
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Convex functions
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Lie groups
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Representations of groups
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Lie-algebra's.
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Holomorfe functies.
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| Form |
Electronic book
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| ISBN |
9783110808148 |
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3110808145 |
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