Finite von Neumann algebras and masas / Allan M. Sinclair, Roger R. Smith.

Format:

Book

Author/Creator:

Sinclair, Allan M.

Contributor:

Smith, Roger R.

Series:

London Mathematical Society lecture note series ; 351.

London Mathematical Society lecture note series ; 351

Language:

English

Subjects (All):

Von Neumann algebras.

Physical Description:

ix, 400 pages : illustrations ; 23 cm.

Place of Publication:

Cambridge [England] ; New York : Cambridge University Press, 2008.

Summary:

Providing a thorough account of the methods that underlie the theory of subalgebras of finite von Neumann algebras, this book contains a substantial amount of current research material and is ideal for those studying operator algebras.

The conditional expectation, basic construction and perturbations within a finite von Neumann algebra with a fixed faithful normal trace are discussed in detail. The general theory of maximal abelian self-adjoint subalgebras (masas) of separable II1 factors is presented with illustrative examples derived from group von Neumann algebras. The theory of singular masas and Sorin Popa's methods of constructing singular and semi-regular masas in a general separable II1 factor and the properties of unbounded operators required for perturbation results.

All proofs are given in considerable detail and standard basic examples are provided, ensuring that the book is accessible to postgraduate students with basic knowledge of von Neumann algebra theory.

Contents:

1 General introduction 1

1.1 Synopsis 1

1.2 Further results 3

2 Masas in B(H) 5

2.1 Introduction 5

2.2 Standard theorems 5

2.3 Masas 8

2.4 Masas in type In algebras 13

3 Finite von Neumann algebras 17

3.1 Introduction 17

3.2 Finite algebras 18

3.3 Examples of masas from groups 21

3.4 Tensor products and crossed products 27

3.5 Diffuse abelian algebras 34

3.6 Conditional expectations 37

3.7 Group von Neumann algebras revisited 48

3.8 Hyperfiniteness 49

4 The basic construction 52

4.1 Introduction 52

4.2 Properties 53

4.3 The trace on (N, eB) 56

4.4 Examples 73

4.5 The pull-down map 76

5 Projections and partial isometries 80

5.1 Introduction 80

5.2 Comparison of two projections 80

5.3 Approximations of projections 88

5.4 Commutants of compressions 91

5.5 Basic lemmas for Kadison's results 93

5.6 Range of the centre-valued trace 95

6 Normalisers, orthogonality and distances 98

6.1 Introduction 98

6.2 Normalisers of masas 98

6.3 Orthogonality of von Neumann subalgebras 104

6.4 Distances between subalgebras 106

7 The Pukánszky invariant 113

7.1 Introduction 113

7.2 The algebras A, A, A' and N(A)" interact 116

7.3 Properties of the Pukánszky invariant 120

7.4 The Pukánszky invariant in group factors 123

7.5 Examples of the Pukánszky invariant 130

7.6 Open problems 136

8 Operators in L∞[0,1] $$B(H) 137

8.1 Introduction 137

8.2 Matrix computations 137

8.3 Main results 141

9 Perturbations 148

9.1 Introduction 148

9.2 Averaging eB over A 150

9.3 Perturbing subalgebras in the uniform norm 156

9.4 Lemmas on close subalgebras 159

9.5 Distances and groupoid normalisers 174

9.6 Numerical constants for perturbations 176

9.7 Perturbations of masas by averaging 183

10 General perturbations 186

10.1 Introduction 186

10.2 The Jones index 186

10.3 Containment of finite algebras 189

10.4 Close von Neumann algebras 193

11 Singular masas 198

11.1 Introduction 198

11.2 Basic lemmas 200

11.3 Singular to WAHP 203

11.4 A basis condition for singularity 208

11.5 Enumeration of words in F2 212

11.6 The Laplacian masa 218

12 Existence of special masas 223

12.1 Introduction 223

12.2 Approximations in subalgebras 224

12.3 Constructing semiregular masas 229

12.4 Constructing singular masas 232

12.5 Singularity and automorphisms 239

13 Irreducible hyperfinite subfactors 242

13.1 Introduction 242

13.2 Irreducible hyperfinite subfactors exist 242

13.3 Cartan masas in hyperfinite subfactors 246

13.4 Property Γ 248

13.5 Irreducible hyperfinites in Γ factors 253

14 Maximal injective subalgebras 257

14.1 Introduction 257

14.2 Maximal injectivity and masas 258

14.3 Maximal injectivity of subfactors 263

15 Masas in non-separable factors 268

15.1 Introduction 268

15.2 Masas in Nω 268

15.3 Masas in L(Fs) 275

16 Singly generated II1 factors 278

16.1 Introduction 278

16.2 Notation and definitions 279

16.3 Examples and basic lemmas 283

16.4 The scaling formula for G 289

16.5 Interpolated free group factors and G 296

16.6 Single generation 298

16.7 Main technical lemmas 302

16.8 Examples of singly generated II1 factors 311

A The ultrapower and property Γ 316

A.1 Introduction 316

A.2 Ultrafilters and characters 317

A.3 Maximal quotients of finite algebras 319

A.4 The algebra Nωn 325

A.5 The ultrapower Nω 328

A.6 Relative commutants in Nω 334

A.7 Property Γ revisited 337

B Unbounded operators 342

B.1 Introduction 342

B.2 Basic results 342

B.3 The functional calculus 348

B.4 Operators from L2(N) 359

B.5 Operators from L1(N) 362

C The trace revisited 373

C.1 Introduction 373

C.2 Preliminary lemmas 373

C.3 Construction of the trace 375.

Notes:

Includes bibliographical references (pages 379-393) and indexes.

ISBN:

9780521719193

0521719194

OCLC:

183926526

Online:

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