Perturbations of subalgebras of type II1 factors

In this paper we consider two von Neumann subalgebras B0 and B of a type II1 factor N. For a map φ on N, we define||φ||∞,2=sup {||φ(x)||2 : ||x||⩽1},and we measure the distance between B0 and B by the quantity ||EB0−EB||∞,2. Under the hypothesis that the relative commutant in 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 ||•||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 A is a masa and u∈N is a unitary such that A and uAu∗ are close, then u must be close to a unitary which normalizes A. These qualitative statements are given quantitative formulations in the paper.