# Asymptotic Orthogonalization of Subalgebras in II$_{1}$ Factors

### Sorin Popa

University of California Los Angeles, USA

## Abstract

Let $M$ be a II$_{1}$ factor with a von Neumann subalgebra $Q⊂M$ that has infinite index under any projection in $Q_{′}∩M$ (e.g., if $Q_{′}∩M$ 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 $M_{ω}$, for a nonprincipal ultrafilter $ω$ on $N$, there exists a unitary element $u∈M_{ω}$ such that $uBu_{∗}$ is orthogonal to $Q_{ω}$.

## Cite this article

Sorin Popa, Asymptotic Orthogonalization of Subalgebras in II$_{1}$ Factors. Publ. Res. Inst. Math. Sci. 55 (2019), no. 4, pp. 795–809

DOI 10.4171/PRIMS/55-4-5