On maximal injective subalgebras

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 $\otimes_{i=1}^\infty\M_i$. 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 $\otimes_{i=1}^\infty\mathcal{L}_{\mathbb{F}_{n_i}}$.