# Asymptotic Orthogonalization of Subalgebras in II$_1$ Factors

### Sorin Popa

University of California Los Angeles, USA

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## 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 $Q^\omega$.

## 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