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

We consider two von Neumann subalgebras $\text{cl}{B}_0$ and $\text{cl}B$ of a type ${\mathrm{I}\mathrm{I}}_1$ factor $\text{cl}N$. For a map $\varphi$ on $\text{cl}N$, we define $$\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{\mathrm{\infty },2}=\sup \{\mathrm{\parallel }\varphi (x){\mathrm{\parallel }}_2\text{\hspace{0.17em}⁣}:\mathrm{\parallel }x\mathrm{\parallel }\le 1\},$$ and we measure the distance between $\text{cl}{B}_0$ and $\text{cl}B$ by the quantity $|{\bb E}{\cl B_0}-{\bb E}{\cl B}|_{\infty,2}$. Under the hypothesis that the relative commutant in $\text{cl}N$ 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 $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2$–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 $\text{cl}A$ is a masa and $u\in \text{cl}N$ is a unitary such that $\text{cl}A$ and $u\text{cl}A{u}^{\ast }$ are close, then $u$ must be close to a unitary which normalizes $\text{cl}A$. These qualitative statements are given quantitative formulations in the paper.