Unitary conjugacy for type III subfactors and W${}^{\ast }$-superrigidity

Abstract

Let $A,B\subset M$ be inclusions of $\sigma$-finite von Neumann algebras such that $A$ and $B$ are images of faithful normal conditional expectations. In this article, we investigate Popa's intertwining condition $A{⪯}_{M}B$ using modular actions on $A$, $B$, and $M$. In the main theorem, we prove that if $A{⪯}_{M}B$, then an intertwining element for $A{⪯}_{M}B$ also intertwines some modular flows of $A$ and $B$. As a result, we deduce a new characterization of $A{⪯}_{M}B$ in terms of the continuous cores of $A$, $B$, and $M$. Using this new characterization, we prove the first W${}^{\ast }$-superrigidity type result for group actions on amenable factors. As another application, we characterize stable strong solidity for free product factors in terms of their free product components.