Unitary conjugacy for type III subfactors and W*-superrigidity

Abstract

Let A, B ⊂ M be inclusions of σ-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*-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.