Theory of Operator Algebras IIIto the Encyclopaedia Subseries on Operator Algebras and Non-Commutative Geometry The theory of von Neumann algebras was initiated in a series of papers by Murray and von Neumann in the 1930's and 1940's. A von Neumann algebra is a self-adjoint unital subalgebra M of the algebra of bounded operators of a Hilbert space which is closed in the weak operator topology. According to von Neumann's bicommutant theorem, M is closed in the weak operator topology if and only if it is equal to the commutant of its commutant. A factor is a von Neumann algebra with trivial centre and the work of Murray and von Neumann contained a reduction of all von Neumann algebras to factors and a classification of factors into types I, II and III. C* -algebras are self-adjoint operator algebras on Hilbert space which are closed in the norm topology. Their study was begun in the work of Gelfand and Naimark who showed that such algebras can be characterized abstractly as involutive Banach algebras, satisfying an algebraic relation connecting the norm and the involution. They also obtained the fundamental result that a commutative unital C* -algebra is isomorphic to the algebra of complex valued continuous functions on a compact space - its spectrum. Since then the subject of operator algebras has evolved into a huge mathematical endeavour interacting with almost every branch of mathematics and several areas of theoretical physics. |
Contents
Ergodic Transformation Groups and the Associated von Neumann Algebras | 1 |
2 Kriegers Construction and Orbit Structure | 14 |
3 Approximately Finite Measured Groupoids | 31 |
4 Amenable Groups and Groupoids | 60 |
Approximately Finite Dimensional von Neumann Algebras | 81 |
1 Inductive Limit and Infinite Tensor Products | 82 |
2 Uniqueness of Approximately Finite Dimensional Factors of Type II₁ | 95 |
3 The Group von Neumann Algebras of Free Groups | 108 |
NonCommutative Ergodic Theory | 252 |
2 Stability of Outer Conjugacy Classes | 260 |
3 Outer Conjugacy of Approximately Inner Automorphisms of Strongly Stable Factors | 270 |
Structure of Approximately Finite Dimensional Factors | 296 |
2 The Flow of Weights and AFD Factors of Type III₀ | 314 |
3 Asymptotic Centralizer | 329 |
4 AFD Factors of Type III₁ | 360 |
Subfactors of an Approximately Finite Dimensional Factor of Type II₁ | 412 |
4 Strongly Stable Factors | 115 |
5 Maximal Abelian Subalgebras | 137 |
Nuclear CAlgebras | 153 |
1 Completely Positive Approximation and Nuclear CAlgebras | 154 |
2 Completely Positive Lifting | 168 |
3 Nuclear CAlgebras and Injective von Neumann Algebras | 173 |
4 GrothendieckHaagerupPisier Inequality | 186 |
Injective von Neumann Algebras | 205 |
2 Finite Injective von Neumann Algebras Second Approach | 237 |
1 AFAlgebras | 413 |
2 Index of Subfactors | 420 |
3 Construction of Subfactors | 441 |
4 Classification of Subfactors of Approximately Finite Dimensional Factors of Type II₁ with Finite Index and Depth | 465 |
Appendix | 495 |
| 523 | |
| 541 | |
| 543 | |
Common terms and phrases
Ad(u Ad(v AFD factor approximately finite dimensional Aut(M Aut(R automorphism group Borel C*-algebra Chapter choose cocycle commute compact completely positive map conditional expectation conjugacy conjugate converges convex countable defined denote dense ei,j equivalent ergodic exists factor of type faithful normal fixed function groupoid Hence Hilbert space homomorphism implies increasing sequence inductive inequality infinite tensor product Int(M Int(R invariant isomorphism L²(M last lemma Let G M₁ matrix unit maximal abelian means Neumann algebra Neumann subalgebra non-zero norm operator algebras orbitally discrete orthogonal pair partial isometry partition Proj(M projection PROOF OF THEOREM Proposition prove Q.E.D. Lemma representation satisfies self-adjoint self-adjoint operator Show strongly central sequence subalgebra subfactor subset Suppose theory topology trace tracial transformation type II1 unitary vector von Neumann algebra