Cartan subalgebras and bimodule decompositions of $ \mathrm{II}_1 $ factors

Abstract

Let $A\subset M$ be a MASA in a $\mathrm{II}_{1}$ factor $M$. We describe the von Neumann subalgebra of $M$ generated by $A$ and its normalizer $\mathcal N(A)$ as the set $N_q^w(A)$ consisting of those elements $m\in M$ for which the bimodule $\smash{\overline{AmA}}$ is discrete. We prove that two MASAs $A$ and $B$ are conjugate by a unitary $u\in N^{w}_{q}(A)$ 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 $B\subset M$, $B=uAu^{*}$ for some $u\in M$ exactly when $A$ is discrete over $B$ and $B$ is discrete over $A$.