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 H 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 II(1) factor with crossed products of abelian algebras by suitably chosen discrete groups.

The spatial isomorphism problem for close separable nuclear C*-algebras.

The Kadison-Kastler problem asks whether close C*-algebras on a Hilbert space must be spatially isomorphic. We establish this when one of the algebras is separable and nuclear. We also apply our methods to the study of near inclusions of C*-algebras.

Property gamma factors and the Hochschild cohomology problem.

The main result of this article is that the kth continuous Hochschild cohomology groups H(k)(M,M ) and H(k)(M, B(H)) of a von Neumann factor subset, dbl equals B(H) of type II(1) with property Gamma are 0 for all positive integers k. The method of proof involves the construction of hyperfinite subfactors with special properties and a new inequality of Grothendieck type for multilinear maps. We prove joint continuity in the parallel. parallel (2) norm of separately ultraweakly continuous multilinear maps and combine these results to reduce to the case of completely bounded cohomology, which is already solved.