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 $Q^{\prime }\cap M$ (e.g., if $Q^{\prime }\cap M$ 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 $uB{u}^{\ast }$ is orthogonal to $Q^{\omega }$.