JournalsjemsVol. 24, No. 5pp. 1679–1721

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

  • Yusuke Isono

    Kyoto University, Japan
Unitary conjugacy for type III subfactors and W$^*$-superrigidity cover
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Let A,BMA,B\subset M be inclusions of σ\sigma-finite von Neumann algebras such that AA and BB are images of faithful normal conditional expectations. In this article, we investigate Popa's intertwining condition AMBA\preceq_MB using modular actions on AA, BB, and MM. In the main theorem, we prove that if AMBA\preceq_MB, then an intertwining element for AMBA\preceq_MB also intertwines some modular flows of AA and BB. As a result, we deduce a new characterization of AMBA\preceq_MB in terms of the continuous cores of AA, BB, and MM. 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.

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Yusuke Isono, Unitary conjugacy for type III subfactors and W^*-superrigidity. J. Eur. Math. Soc. 24 (2022), no. 5, pp. 1679–1721

DOI 10.4171/JEMS/1135