Perturbations of subalgebras of type II1 factors Academic Article uri icon

abstract

  • 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 u A u* are close, then u must be close to a unitary which normalizes A. These qualitative statements are given quantitative formulations in the paper. © 2004 Elsevier Inc. All rights reserved.

author list (cited authors)

  • Popa, S., Sinclair, A. M., & Smith, R. R.

citation count

  • 8

complete list of authors

  • Popa, Sorin||Sinclair, Allan M||Smith, Roger R

publication date

  • August 2004