Cartan Subalgebras and Bimodule Decompositions of Ii1 Factors

Let A ⊂M be a MASA in a II1 factor M. We describe the von Neumann subalgebra of M generated by A and its normalizer N (A) as the set Nw q (A) consisting of those elements m ∈M for which the bimodule AmA is discrete. We prove that two MASAs A and B are conjugate by a unitary u ∈ Nw q (A) iff A is discrete over B and B is discrete over A in the sense defined by Feldman and Moore [5]. As a consequence, we show that A is a Cartan subalgebra of M iff for any MASA B⊂M, B = uAu∗ for some u ∈M exactly when A is discrete over B and B is discrete over A.