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

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 $\hat{\mathbb{G}}$.