Linear Analysis and Representation Theory Book

Chapter I Algebras and Banach Algebras 1

1 Algebras and Norms 1

2 The Group of Units and the Quasigroup 5

3 The Maximal Ideal Space 8

4 The Spectrum of an Element 10

5 The Spectral Norm Formula 13

6 Commutative Banach Algebras and their Ideals 16

7 Radical and Semisimplicity 26

8 Involutive Algebras 37

9 H^* Algebras 45

Remarks 52

Chapter II Operators and Operator Algebras 55

1 Topologies on Vector Spaces and on Operator Algebras 55

2 Compact Operators 65

3 The Spectral Theorem for Compact Operators 68

4 Hubert-Schmidt Operators 71

5 Trace Class Operators 76

6 Vector Valued Line Integrals 82

7 Homomorphisms into A. The Spectral Mapping Theorem 85

8 Unbounded Operators 90

Remarks 99

Chapter III The Spectral Theorem, Stable Subspaces and v. Neumann Algebras 102

1 Linear Functionals on Vector Lattices and their Extensions 102

2 Linear Functionals on Lattices of Functions 108

3 The Spectral Theorem for Self Adjoint Operators in Hilbert Space 111

4 Normal Elements and Normal Operators 116

5 Stable Subspaces and Commutants 122

6 von Neumann Algebras 127

7 Measures on Locally Compact Spaces 135

Remarks 143

Chapter IV Elementary Representation Theory in Hilbert Space 145

1 Representations and Morphisms 145

2 Irreducible Components, Equivalence 150

3 Intertwining Operators 158

4 Schur's Lemma 161

5 Multiplicity of Irreducible Components 168

6 The General Trace Formula 175

7 Primary Representations and Factorial v. Neumann Algebras 183

8 Algebras and Representations of Type I 198

9 Type II and III v. Neumann Algebras 211

Remarks 225

Preliminary Remarks to Chapter V 227

Chapter V Topological Groups, Invariant Measures, Convolutions and Representations 228

1 Topological Groups and Homogeneous Spaces 228

2 Haar Measure 241

3 Quasi-Invariant and Relatively Invariant Measures 257

4 Convolutions of Functions and Measures 270

5 The Algebra Representation Associated with ρ: S→L (H) 284

6 The Regular Representations of Locally Compact Groups 299

7 Continuity of Group Representations and the Gelfand-Raikov Theorem 303

Remarks 318

Chapter VI Induced Representations 321

1 The Riesz-Fischer Theorem 321

2 Induced Representations when G/H has an Invariant Measure 325

3 Tensor Products 332

4 Induced Representations for Arbitrary G and H 347

5 The Existence of a Kernel for δ: L¹(G)→L(K) 360

6 The Direct Sum Decomposition of the Induced Representation ρ^x: G→u (K) 367

7 The Isometric Isomorphism between L² and HS(K₁, K¹). The Computation of the Trace in Terms of the Associated Kernel 376

8 The Tensor Product of Induced Representations 387

9 The Theorem on Induction in Stages 393

10 Representations Induced by Representations of Conjugate Subgroups 398

11 Mackey's Theorem on Strong Intertwining Numbers and Some of its Consequences 402

12 Isomorphism Theorems Implying the Frobenius Reciprocity Relation 411

Remarks 419

Chapter VII Square Integrable Representations, Spherical Functions and Trace Formulas 423

1 Square Integrable Representations and the Representation Theory of Compact Groups 423

2 Zonal Spherical Functions 444

3 Spherical Functions of Arbitrary Type and Height 460

4 Godement's Theorem on the Characterization of Spherical Functions 477

5 Representations of Groups with an Iwasawa Decomposition 494

6 Trace Formulas 510

Remarks 526

Chapter VIII Lie Algebras, Manifolds and Lie Groups 531

1 Lie Algebras 531

2 Finite Dimensional Representations of Lie Algebras. Cartan's Criteria and the Theorems of Engel and Lie 546

3 Presheaves and Sheaves 560

4 Differentiable Manifolds 573

5 Lie Groups and their Lie Algebras 588

6 The Exponential Map and Canonical Coordinates 608

7 Lie Subgroups and Subalgebras 629

8 Invariant Lie Subgroups and Quotients of Lie Groups. The Projective Groups and the Lorentz Group 644

Remarks 659

Bibliography 663

Subject Index 672

Index of Notations and Special Symbols 682