Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

Isono, Yusuke

doi:10.2140/apde.2019.12.1295

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Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

Yusuke Isono

Vol. 12 (2019), No. 5, 1295–1324

DOI: 10.2140/apde.2019.12.1295

Abstract

Let $G$ be a free (unitary or orthogonal) quantum group. We prove that for any nonamenable subfactor $N \subset L^{\infty} (G)$ which is an image of a faithful normal conditional expectation, and for any $σ$ -finite factor $B$ , the tensor product $N \bar{\otimes} B$ has no Cartan subalgebras. This generalizes our previous work that provides the same result when $B$ is finite. In the proof, we establish Ozawa–Popa and Popa–Vaes’s weakly compact action on the continuous core of $L^{\infty} (G) \bar{\otimes} B$ as the one relative to $B$ , by using an operator-valued weight to $B$ and the central weak amenability of $\hat{G}$ .

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Keywords

von Neumann algebra, type III factor, Cartan subalgebra

Mathematical Subject Classification 2010

Primary: 46L10, 46L36

Secondary: 58B32

Milestones

Received: 11 January 2018

Accepted: 16 September 2018

Published: 15 December 2018

Authors

Yusuke Isono
Research Institute for Mathematical Sciences Kyoto University Kyoto Japan

Vol. 12, No. 5, 2019