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

Abstract

We consider two von Neumann subalgebras $\cl B_0$ and $\cl B$ of a type ${\rm{II}}_1$ factor $\cl N$. For a map $ϕ$ 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 $\|\cdot \|_2$--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.