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Finite von Neumann Algebras and Masas

By: Allan Sinclair, Roger Smith

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Paperback | 26 June 2008

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412 Pages

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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 general separable II1 factor are explored. Appendices cover the ultrapower of an II1 factor and the properties of unbounded operators required for perturbation results. Proofs are given in considerable detail and standard basic examples are provided, making the book understandable to postgraduates with basic knowledge of von Neumann algebra theory.

Industry Reviews

'Sinclair and Smith's monograph is very well written ... well suited for graduate students who have been given a first course on operator algebras, for Ph.D. students who have started working on finite von Neumann algebras, but also for specialists because it gathers much useful and technical material.' Mathematical Reviews
'... suitable for graduate students wanting to learn this part of mathematics.' EMS Newsletter

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Non-Fiction Mathematics Algebra Calculus & Mathematical Analysis Differential Calculus & Equations

Preface	p. ix
General introduction	p. 1
Synopsis	p. 1
Further results	p. 3
Masas in B(H)	p. 5
Introduction	p. 5
Standard theorems	p. 5
Masas	p. 8
Masas in type I_n algebras	p. 13
Finite von Neumann algebras	p. 17
Introduction	p. 17
Finite algebras	p. 18
Examples of masas from groups	p. 21
Tensor products and crossed products	p. 27
Diffuse abelian algebras	p. 34
Conditional expectations	p. 37
Group von Neumann algebras revisited	p. 48
Hyperfiniteness	p. 49
The basic construction	p. 52
Introduction	p. 52
Properties	p. 53
The trace on (N, eB)	p. 56
Examples	p. 73
The pull-down map	p. 76
Projections and partial isometries	p. 80
Introduction	p. 80
Comparison of two projections	p. 80
Approximations of projections	p. 88
Commutants of compressions	p. 91
Basic lemmas for Kadison's results	p. 93
Range of the centre-valued trace	p. 95
Normalisers, orthogonality and distances	p. 98
Introduction	p. 98
Normalisers of masas	p. 98
Orthogonality of von Neumann subalgebras	p. 104
Distances between subalgebras	p. 106
The Pukánszky invariant	p. 113
Introduction	p. 113
The algebras A, A, A' and N(A)" interact	p. 116
Properties of the Pukánszky invariant	p. 120
The Pukánszky invariant in group factors	p. 123
Examples of the Pukánszky invariant	p. 130
Open problems	p. 136
Operators in <$$$>	p. 137
Introduction	p. 137
Matrix computations	p. 137
Main results	p. 141
Perturbations	p. 148
Introduction	p. 148
Averaging e_B over A	p. 150
Perturbing subalgebras in the uniform norm	p. 156
Lemmas on close subalgebras	p. 159
Distances and groupoid normalisers	p. 174
Numerical constants for perturbations	p. 176
Perturbations of masas by averaging	p. 183
General perturbations	p. 186
Introduction	p. 186
The Jones index	p. 186
Containment of finite algebras	p. 189
Close von Neumann algebras	p. 193
Singular masas	p. 198
Introduction	p. 198
Basic lemmas	p. 200
Singular to WAHP	p. 203
A basis condition for singularity	p. 208
Enumeration of words in F₂	p. 212
The Laplacian masa	p. 218
Existence of special masas	p. 223
Introduction	p. 223
Approximations in subalgebras	p. 224
Constructing semiregular masas	p. 229
Constructing singular masas	p. 232
Singularity and automorphisms	p. 239
Irreducible hyperfinite subfactors	p. 242
Introduction	p. 242
Irreducible hyperfinite subfactors exist	p. 242
Cartan masas in hyperfinite subfactors	p. 246
Property	p. 248
Irreducible hyperfinites in factors	p. 253
Maximal injective subalgebras	p. 257
Introduction	p. 257
Maximal injectivity and masas	p. 258
Maximal injectivity of subfactors	p. 263
Masas in non-separable factors	p. 268
Introduction	p. 268
Masas in N	p. 268
Masas in L(F_s)	p. 275
Singly generated II₁ factors	p. 278
Introduction	p. 278
Notation and definitions	p. 279
Examples and basic lemmas	p. 283
The scaling formula for G	p. 289
Interpolated free group factors and G	p. 296
Single generation	p. 298
Main technical lemmas	p. 302
Examples of singly generated II₁ factors	p. 311
The ultrapower and property	p. 316
Introduction	p. 316
Ultrafilters and characters	p. 317
Maximal quotients of finite algebras	p. 319
The algebra N_n	p. 325
The ultrapower N	p. 328
Relative commutants in N	p. 334
Property revisited	p. 337
Unbounded operators	p. 342
Introduction	p. 342
Basic results	p. 342
The functional calculus	p. 348
Operators from L²(N)	p. 359
Operators from L¹(N)	p. 362
The trace revisited	p. 373
Introduction	p. 373
Preliminary lemmas	p. 373
Construction of the trace	p. 375
Bibliography	p. 379
Index	p. 394
Index of symbols	p. 398
Table of Contents provided by Ingram. All Rights Reserved.

Finite von Neumann Algebras and Masas

At a Glance

Paperback

Industry Reviews

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