Maximal Injective Subalgebras of Tensor Products of Free Groups Factors

Abstract

In this article, we proved the following results. Let $L(F(n_i))$ be the free group factor on $n_i$ generators and $\lambda (g_i)$ be one of standard generators of $L(F(n_i))$ for $1\le i\le N$. Let $\A_i$ be the abelian von Neumann subalgebra of $L(F(n_i))$ generated by $\lambda (g_i)$. Then the abelian von Neumann subalgebra $\otimes_{i=1}^N\A_i$ is a maximal injective von Neumann subalgebra of ${\otimes }_{i=1}^{N}L(F(n_i))$. When $N$ is equal to infinity, we obtained McDuff factors that contain maximal injective abelian von Neumann subalgebras.