# Unitary conjugacy for type III subfactors and W$^*$-superrigidity

## TL;DR

This paper explores the relationship between subalgebra inclusions and modular actions in von Neumann algebras, leading to new characterizations and the first superrigidity results for certain group actions.

## Contribution

It introduces a novel characterization of Popa's intertwining condition via modular flows and applies it to establish W$^*$-superrigidity for group actions on amenable factors.

## Key findings

- New characterization of intertwining in terms of modular flows
- First W$^*$-superrigidity result for group actions on amenable factors
- Characterization of stable strong solidity for free product factors

## 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 $A\preceq_MB$ using their modular actions. In the main theorem, we prove that if $A\preceq_MB$ holds, then an intertwining element for $A\preceq_MB$ also intertwines some modular flows of $A$ and $B$. As a result, we deduce a new characterization of $A\preceq_MB$ in terms of their continuous cores. 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.