Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles (eBook)

Basic Representation Theory of Groups and Algebras

R. S. Doran, J. M.G. Fell (Herausgeber)

eBook Download: EPUB

1988 | 1. Auflage
746 Seiten
Elsevier Science (Verlag)
978-0-08-087444-9 (ISBN)

Infos

This is an all-encompassing and exhaustive exposition of the theory of infinite-dimensional Unitary Representations of Locally Compact Groups and its generalization to representations of Banach algebras. The presentation is detailed, accessible, and self-contained (except for some elementary knowledge in algebra, topology, and abstract measure theory). In the later chapters the reader is brought to the frontiers of present-day knowledge in the area of Mackey normal subgroup analysisand its generalization to the context of Banach *-Algebraic Bundles.

Inhalt

Front Cover 1
Basic Representation Theory of Groups and Algebras 4
Copyright Page 5
Contents 8
Preface 14
Introduction to Volume 1 (Capter I to VII) 20
Chapter I Preliminaries 60
1. Logical, Set and Functional Notation 60
2. Numerical Notation 61
3. Topology 63
4. Algebra 65
5. Seminorms, Normed Spaces, Normed Algebras 68
6. Hilbert Spaces 72
7. Exercises for Chapter I 76
Notes and Remarks 80
Chapter II Integration Theory and Banach Bundles 82
1. d-Rings, Measures, and Measurable Functions 84
2. Integration of Complex Functions 88
3. The Outsize Lp Spaces 93
4. Local Measurability Structures 95
5. Integration of Functions with Values in a Banach Space 100
6. Integration of Functions with Values in a Locally Convex Space 104
7. The Radon-Nikodym Theorem and Related Topics 106
8. Measures on Locally Compact Hausdorff Spaces 110
9. Product Measures and Fubini’s Theorem 117
10. Measure Transformations 126
11. Projection-Valued Measures and Spectral Integrals 131
12. The Analogue of the Riesz Theorem for Projection-Valued Measures 139
13. Banach Bundles 144
14. Banach Bundles over Locally Compact Base Spaces' 155
15. Integration in Banach Bundles over Locally Compact Spaces 164
16. Fubini Theorems for Banach Bundles 173
17. Exercises for Chapter II 176
Notes and Remarks 180
Chapter III Locally Compact Groups 182
1. Topological Groups and Subgroups 183
2. Quotient Spaces and Homomorphisms 190
3. Topological Transformation Spaces 194
4. Direct and Semidirect Products 200
5. Group Extensions 205
6. Topological Fields 214
7. Haar Measure 219
8. The Modular Function 227
9. Examples of Haar Measure and the Modular Function 232
10. Convolution and Involution of Measures on G 237
11. Convolution of functions, the L1, group algebra 244
12. Relations between Measure and Topology on G 253
13. Invariant Measures on Coset Spaces 255
14. Quasi-Invariant Measures on Coset Spaces 265
15. Exercises for Chapter III 276
Notes and Remarks 281
Chapter IV Algebraic Representation Theory 284
1. Fundamental Definitions 285
2. Complete Reducibility and Multiplicity for Operator Sets 288
3. Representations of Groups and Algebras 295
4. The Extended Jacobson Density Theorem 302
5. Finite-dimensional Semisimple Algebras 309
6. Application to Finite Groups 319
7. The Complex Field and *-Algebras 327
8. Exercises for Chapter IV 334
Notes and Remarks 340
Chapter V Locally Convex Representations and Banach Algebras 342
1. Locally Convex Representations Fundamental Definitions
2. Extending Locally Convex Representations of Two-sided Ideals 350
3. The Naimark Relation 352
4. Elementary Remarks on Normed Algebras Examples
5. The Spectrum 359
6. Spectra in Banach Algebras, Mazur's Theorem, Gelfand’s Theorem 365
7. Commutative Banach Algebras 373
8. Function Algebras and Co(S) 377
9. Factorization in Banach Algebras 384
10. Exercises for Chapter V 387
Notes and Remarks 393
Chapter VI C*-Algebras and Their *-Representations 396
1. *-Algebras Elementary Remarks and Examples
2. Symmetric *-Algebra 401
3. C*-Algebras 404
4. Commutative C*-Algebras 409
5. Spectra in Subalgebras of C*-Algebras 411
6. The Functional Calculus in C*-Algebras 413
7. Positive Elements and Symmetry of C*-Algebras 415
8. Approximate Units in a C*-Algebra Applications to Ideals and Quotients
9. Elementary Remarks on *-Representations 432
10. The C*-Completion of a Banach *-Algebra Stone’s Theorem
11. The Spectral Theory of Bounded Normal Operators 442
12. The Spectral Theory of Unbounded Normal Operators 450
13. Polar Decomposition of Operators Mackey’s Form of Schur’s Lemma
14. A Criterion for Irreducibility Discrete Multiplicity Theory
15. Compact Operators and Hilbert-Schmidt Operators 469
16. The Sturm-Liouville Theory 480
17. Inductive Limits of C*-Algebras 487
18. Positive Functionals 489
19. Positive Functionals and *-Representations 495
20. Indecomposable Positive Functionals and Irreducible *-Representations 504
21. Positive Functionals on Commutative *-Algebras the Generalized Bochner and Plancherel Theorems
22. The Existence of Positive Functionals and *-Representations of C*-Algebras the Gelfand–Naimark Theorem
23. Application of Extension Techniques to the Algebra of Compact Operators 524
24. Von Neumann Algebras and *-Algebras with Type I Representation Theory 527
25. Kadison’s Irreducibility Theorem and Related Properties of C*-Algebras 533
26. Exercises for Chapter VI 541
Notes and Remarks 550
Chapter VII The Topology of the Space of *-Representations 558
1. The Definition and Elementary Properties of the Regional Topology 560
2. The Regional Topology and Separation Properties 571
3. The Structure Space 575
4. Restriction of Representations to Hereditary Subalgebras 578
5. The Regional and Hull-Kernel Topologies on the Structure Space of a C*-Algebra 582
6. The Baire Property and Local Compactness of Â 592
7. C*-Algebras with Finite Structure Space 596
8. Bundles of C*-Algebras 598
9. The Spectral Measure of a *-Representation 612
10. *-Representations Whose Spectral Measures are Concentrated at a Single Point 621
11. Exercises for Chapter VII 626
Notes and Remarks 629
Appendix A The Stone–Weierstrass Theorems 632
Appendix B Unbounded Operators in Hilbert Space 638
Appendix C The Existence of Continuous Cross-Sections of Banach Bundles 652
Bibliography 662
Name Index 740
Subject Index 744
Index of Notation 758

Produkt-Details

• Erscheint lt. Verlag: 15.4.1988
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Technik
• ISBN-10: 0-08-087444-4 / 0080874444
• ISBN-13: 978-0-08-087444-9 / 9780080874449