Perturbations of Subalgebras of Type ${\mathrm{II}}_1$ Factors

Abstract

We consider two von Neumann subalgebras $\cl B_0$ and $\cl B$ of a type ${\mathrm{II}}_1$ factor $\cl N$. For a map $\varphi$ on $\cl N$, we define \[\|ϕ\|_{\infty,2}=\sup\{\|ϕ(x)\|_2\colon \|x\| \leq 1\},\] and we measure the distance between $\cl B_0$ and $\cl B$ by the quantity $\|{\bb E}_{\cl B_0}-{\bb E}_{\cl B}\|_{\infty,2}$. Under the hypothesis that the relative commutant in $\cl N$ of each algebra is equal to its center, we prove that close subalgebras have large compressions which are spatially isomorphic by a partial isometry close to 1 in the $\Vert \cdot {\Vert }_2$--norm. This hypothesis is satisfied, in particular, by masas and subfactors of trivial relative commutant. A general version with a slightly weaker conclusion is also proved. As a consequence, we show that if $\cl A$ is a masa and $u\in\cl N$ is a unitary such that $\cl A$ and $u\cl Au^*$ are close, then $u$ must be close to a unitary which normalizes $\cl A$. These qualitative statements are given quantitative formulations in the paper.

This submission corrects both an error in the statement of Lemma 2.1(ii) and an induced error in the proof of Theorem 5.2, the statement of which remains the same. These were discovered after publication, so a correction will appear in J. Funct. Anal