Jan Cameron

Kadison-Kastler stable factors

A conjecture of Kadison and Kastler from 1972 asks whether sufficiently close operator algebras in a natural uniform sense must be small unitary perturbations of one another. For n ≥ 3 and a free ergodic probability measure preserving action of SLn(Z) on a standard nonatomic probability space (X,µ), write M = ((L 1 (X,µ)⋊SLn(Z))⊗R, where R is the hyperfinite II1 factor. We show that whenever M is represented as a von Neumann algebra on some Hilbert space H and N ⊆ B(H) is sufficiently close toM, then there is a unitary u on H close to the identity operator with uMu � = N. This provides the first nonamenable class of von Neumann algebras satisfying Kadison and Kastlers conjecture. We also obtain stability results for crossed products L 1 (X,µ)⋊ whenever the compar- ison map from the bounded to usual group cohomology vanishes in degree 2 for the module L 2 (X,µ). In this case, any von Neumann algebra sufficiently close to such a crossed product is necessarily isomorphic to it. In particular, this result applies when is a free group. This paper provides a complete account of the results announced in (12).

Type II1 factors satisfying the spatial isomorphism conjecture

This paper addresses a conjecture in the work by Kadison and Kastler [Kadison RV, Kastler D (1972) Am J Math 94:38–54] that a von Neumann algebra M on a Hilbert space should be unitarily equivalent to each sufficiently close von Neumann algebra N, and, moreover, the implementing unitary can be chosen to be close to the identity operator. This conjecture is known to be true for amenable von Neumann algebras, and in this paper, we describe classes of nonamenable factors for which the conjecture is valid. These classes are based on tensor products of the hyperfinite II1 factor with crossed products of abelian algebras by suitably chosen discrete groups.

Structural properties of close II1 factors

We show that a number of key structural properties transfer between sufficiently close II

Intermediate subalgebras and bimodules for general crossed products of von Neumann algebras

_1

The radial masa in a free group factor is maximal injective

factors, including solidity, strong solidity, uniqueness of Cartan masas and property

A remark on the similarity and perturbation problems

\Gamma

Bimodules over Cartan MASAs in von Neumann Algebras, Norming Algebras, and Mercer's Theorem

. We also examine II

Mixing subalgebras of finite von Neumann algebras

_1

Bimodules in crossed products of von Neumann algebras

factors close to tensor product factors, showing that such factors also factorise as a tensor product in a fashion close to the original.

A Galois correspondence for reduced crossed products of unital simple C

Let G be a discrete group acting on a von Neumann algebra M by properly outer ∗-automorphisms. In this paper, we study the containment M ⊆ M ⋊αG of M inside the crossed product. We characterize the intermediate von Neumann algebras, extending earlier work of other authors in the factor case. We also determine the M-bimodules that are closed in the Bures topology and which coincide with the w∗-closed ones under a mild hypothesis on G. We use these results to obtain a general version of Mercer’s theorem concerning the extension of certain isometric w∗-continuous maps on M-bimodules to ∗-automorphisms of the containing von Neumann algebras.