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. ∗Partially supported by grants from the National Science Foundation.