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.
Jan Cameron, Erik Christensen, Allan M Sinclair, Roger R Smith, Stuart A White, and Alan D Wiggins.
Department of Mathematics, Vassar College, Poughkeepsie, NY 12604, USA.
Proc. Natl. Acad. Sci. U.S.A. 2012 Dec 11; 109 (50): 20338-43.
AbstractThis 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.