#### Vol. 12, No. 5, 2019

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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
##### Abstract

Let $\mathbb{G}$ be a free (unitary or orthogonal) quantum group. We prove that for any nonamenable subfactor $N\subset {L}^{\infty }\left(\mathbb{G}\right)$ which is an image of a faithful normal conditional expectation, and for any $\sigma$-finite factor $B$, the tensor product $N\overline{\otimes }B$ 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 ${L}^{\infty }\left(\mathbb{G}\right)\overline{\otimes }B$ as the one relative to $B$, by using an operator-valued weight to $B$ and the central weak amenability of $\stackrel{̂}{\mathbb{G}}$.

##### Keywords
von Neumann algebra, type III factor, Cartan subalgebra
##### Mathematical Subject Classification 2010
Primary: 46L10, 46L36
Secondary: 58B32
##### Milestones
Accepted: 16 September 2018
Published: 15 December 2018
##### Authors
 Yusuke Isono Research Institute for Mathematical Sciences Kyoto University Kyoto Japan