Asymptotic orthogonalization of subalgebras in II$_1$ factors

TL;DR

This paper proves that in certain II$_1$ factors, any separable subalgebra in the ultrapower can be unitarily orthogonalized relative to a subalgebra with infinite index, revealing asymptotic orthogonalization properties.

Contribution

It establishes the existence of unitaries in ultrapower II$_1$ factors that orthogonalize separable subalgebras relative to subalgebras with infinite index.

Findings

Existence of unitaries orthogonalizing subalgebras in ultrapower II$_1$ factors

Applicable to subalgebras with infinite index, including abelian and irreducible subfactors

Advances understanding of asymptotic orthogonalization in operator algebras

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., $Q$ abelian; or $Q$ 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 non-principal ultrafilter $\omega$ on $\Bbb N$, there exists a unitary element $u\in M^\omega$ such that $uBu^*$ is orthogonal to $Q^\omega$.