Perturbations of subalgebras of type II1 factors

In this paper we consider two von Neumann subalgebras B-0 and B of a type III factor N. For a map phi on N, we define parallel tophiparallel to(infinity2) = sup {parallel tophi(x)parallel to(2):parallel toxparallel toless than or equal to1}, and we measure the distance between B-0 and B by the quantity parallel toEB(0) - E(B)parallel to(infinity,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 I in the parallel to(.)parallel to(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 epsilon 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. (C) 2004 Elsevier Inc. All rights reserved.