A remark on central sequence algebras of the tensor product of $\mathrm {II}_{1}$ factors

Abstract:

Let $\mathcal {M}$ and $\mathcal {N}$ be two type $\mathrm {II}_{1}$ factors with separable predual and $\omega$ a free ultrafilter on $\mathbb {N}$. If the central sequence algebra $\mathcal {N}_{\omega }$ is abelian and there is a non-atomic abelian subalgebra $\mathcal {A}$ in $\mathcal {M}$ such that any central sequence of $\mathcal {M}\overline {\otimes }\mathcal {N}$ is contained in the ultrapower $(\mathcal {A}\overline {\otimes }\mathcal {N})^{\omega }$, then $(\mathcal {M}\overline {\otimes }\mathcal {N})_{\omega }$ is abelian. It is also shown that there is an action $\alpha$ of the free group $F_2$ on the group von Neumann algebra $\mathcal {L}_{\mathbb {Z}}$ such that the central sequence algebra of $\mathcal {M}=\mathcal {L}_{\mathbb {Z}}\rtimes _{\alpha } F_2$ is abelian and non-trivial and any central sequence in $\mathcal {M}\overline {\otimes }\mathcal {N}$ is in the ultrapower $(\mathcal {L}_{\mathbb {Z}}\overline {\otimes }\mathcal {N})^{\omega }$.