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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Abstract

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.

Cite this article

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