A Kurosh Type Theorem for Type Ii1 Factors

The classification of type II1 factors (of discrete groups) was initiated by Murray and von Neumann [MvN] who distinguished the hyperfinite type II1 factor R from the group factor LFr of the free group Fr on r ≥ 2 generators. Thirty years later, Connes [Co2] proved uniqueness of the injective type II1 factor. Thus, the group factor LΓ of an ICC amenable group Γ is isomorphic to the hyperfinite type II1 factorR. On the other hand, the isomorphism problem of free group factors remains open. To solve this problem, Voiculescu invented free probability theory, which led to a number of deep results on the structure of free group factors (cf. the survey paper [Vo2]). Apart from these results and results of Connes [Co1] and Cowling and Haagerup [CH], the classification of type II1 factors has been vague by and large. Recently, however, a breakthrough came when Popa [Po2][Po4] found that unitary conjugacy results can be deduced from existence of finite-dimensional bimodules and obtained quite precise classification theorems for certain classes of type II1 factors. On the other hand, a C-algebraic method [Oz2] was proved to be useful in study of type II1 factors. These methods in combination yielded some prime factorization results in [OP]. This paper is a continuation of [Oz2] and [OP], where the structure of (tensor products of) word hyperbolic group type II1 factors was studied. In this paper, we will study the structure of free-products and crossed products of certain type II1 factors. A crucial ingredient of the argument is a computation of the kernels of certain morphisms on C-algebras. The idea of exploiting a ‘boundary’ to compute such kernels is due to Skandalis [Sk] and developed by Higson and Guentner [HG]. We will take advantage of this idea. We denote by C the class of countable discrete groups Γ such that the left and right translation action of Γ× Γ on its Stone-Čech boundary ∂Γ = βΓ \ Γ is amenable (see Section 4 for details). The class C was suggested by Skandalis and it contains all subgroups of word hyperbolic groups and discrete subgroups of connected simple Lie groups of rank one [HG][Sk]. The class C also contains a group with an infinite amenable normal subgroup (cf. Corollary