Asymptotic Orthogonalization of Subalgebras in II$_1$ Factors

Abstract

Let M be a II _1 factor with a von Neumann subalgebra Q\subset M that has infinite index under any projection in Q'\cap M (e.g., if Q'\cap M is diffuse, or if Q is an irreducible subfactor with infinite Jones index). We prove that given any separable subalgebra B of the ultrapower II _1 factor M^\omega , for a nonprincipal ultrafilter \omega on \mathbb{N} , there exists a unitary element u\in M^\omega such that uBu^* is orthogonal to Q^\omega .