A Kurosh type theorem for type II_1 factors

We prove a Kurosh type theorem for free-product type II_1 factors. In particular, if M = LF_2 \otimes R, then the free-product type II_1 factors M*…*M are all prime and pairwise non-isomorphic. This paper is a continuation of [N. Ozawa, Solid von Neumann algebras] and [N. Ozawa and S. Popa, Some prime factorization results for type II_1 factors].