A remark on central sequence algebras of the tensor product of 𝐼𝐼₁ factors

Abstract

Let M \mathcal {M} and N \mathcal {N} be two type I I 1 \mathrm {II}_{1} factors with separable predual and ω \omega a free ultrafilter on N \mathbb {N} . If the central sequence algebra N ω \mathcal {N}_{\omega } is abelian and there is a non-atomic abelian subalgebra A \mathcal {A} in M \mathcal {M} such that any central sequence of M ⊗ ¯ N \mathcal {M}\overline {\otimes }\mathcal {N} is contained in the ultrapower ( A ⊗ ¯ N ) ω (\mathcal {A}\overline {\otimes }\mathcal {N})^{\omega } , then ( M ⊗ ¯ N ) ω (\mathcal {M}\overline {\otimes }\mathcal {N})_{\omega } is abelian. It is also shown that there is an action α \alpha of the free group F 2 F_2 on the group von Neumann algebra L Z \mathcal {L}_{\mathbb {Z}} such that the central sequence algebra of M = L Z ⋊ α F 2 \mathcal {M}=\mathcal {L}_{\mathbb {Z}}\rtimes _{\alpha } F_2 is abelian and non-trivial and any central sequence in M ⊗ ¯ N \mathcal {M}\overline {\otimes }\mathcal {N} is in the ultrapower ( L Z ⊗ ¯ N ) ω (\mathcal {L}_{\mathbb {Z}}\overline {\otimes }\mathcal {N})^{\omega } .