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

Abstract

Let G be a free (unitary or orthogonal) quantum group. We prove that for any nonamenable subfactor N⊂L∞(G) which is an image of a faithful normal conditional expectation, and for any σ-finite factor B, the tensor product N⊗¯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∞(G)⊗¯B as the one relative to B, by using an operator-valued weight to B and the central weak amenability of G .