Free group factors and factors with some decompositions

Abstract

In this paper, we show that if type $II_1$ von Neumann factors $\mathcal {M}$ have some decompositions introduced by Liming Ge and Sorin Popa, then these von Neumann factors are not isomorphic to free group factors $L(F_n)\ (n \geq 2)$. Thus we have proved the number $l_a$ defined by Ge and Popa bigger than 3 for all free group factors and we also extend some results of M. Stefan.