Type II_1 factors satisfying the spatial isomorphism conjecture

TL;DR

This paper proves the Kadison-Kastler conjecture for new classes of non-amenable type II_1 factors, showing they are stable under small perturbations in the operator algebra setting.

Contribution

It extends the validity of the spatial isomorphism conjecture to tensor products involving hyperfinite II_1 factors and crossed products by discrete groups.

Findings

Kadison-Kastler conjecture holds for these new classes

Non-amenable factors are stable under small perturbations

Tensor product constructions preserve the conjecture

Abstract

This paper addresses a conjecture of Kadison and Kastler 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 is known to be true for amenable von Neumann algebras and in this paper we describe new classes of non-amenable factors for which the conjecture is valid. These are based on tensor products of the hyperfinite II_1 factor with crossed products of abelian algebras by suitably chosen discrete groups.