On maximal injective subalgebras

Gao, Mingchu

doi:10.1090/S0002-9939-10-10219-6

On maximal injective subalgebras
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Proc. Amer. Math. Soc. 138 (2010), 2065-2070 Request permission

Abstract:

Let $\mathcal {A}_i$ be a type $I$ von Neumann subalgebra in a type $II_1$ factor $\mathcal {M}_i$ with the faithful trace $\tau _i$ such that $\mathcal {A}_i’\cap \mathcal {M}_i\subseteq \mathcal {A}_i$, for $i=1, 2, \cdots$. Moreover, suppose $\mathcal {A}_i$ has the asymptotically orthogonal property in $\mathcal {M}_i$ after tensoring the finite von Neumann algebra $\otimes _{j\ne i}\mathcal {M}_j$, for all $i=1,2,\cdots$. Then we show that $\otimes _{i=1}^\infty \mathcal {A}_i$ is maximal injective in the infinite tensor product von Neumann algebra $\otimes _{i=1}^\infty \mathcal M_i$. As a consequence, we get the following result. Let $\{\mathbb {F}_{n_i};i=1,2, \cdots \}$ be a sequence of free groups with $n_i$ ($>1$) generators. Let $\mathcal {A}_i$ be the masa of group von Neumann algebra $\mathcal {L}_{\mathbb {F}_{n_i}}$ generated by a generator of $\mathbb {F}_{n_i}$ or by the sum of all generators and their inverses of the group. Then $\otimes _{i=1}^\infty \mathcal {A}_i$ is maximal injective in the infinite tensor product von Neumann algebra $\otimes _{i=1}^\infty \mathcal {L}_{\mathbb {F}_{n_i}}$.

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