A Kurosh-type Theorem for Type Iii Factors

In [Oz], Ozawa obtained analogues of the Kurosh subgroup theorem (and its consequences) in the setting of free products of semiexact II1 factors with respect to the canonical tracial states. In particular he was able to prove a certain unique-factorization theorem that distinquishes, for example, the n-various L(F∞) ∗ (L(F∞)⊗ R) . The paper was a continuation of the joint work [OP] with Popa that proved various unique-factorization results for tensor products of II1 factors. These papers shared a particular combination of ideas from [Oz2] and [Po3]. The first was a C-algebraic method for detecting injectivity of a von Neumann algebra, adopted from its original context of proving solidity of finite von Neumann algebras with the AkemannOstrand property. This method was used in concert with Popa’s intertwining-by-bimodules technique that (roughly speaking) allows one to conclude unitary conjugacy results from a spatial condition. More precisely, if A and B are diffuse subalgebras of a finite von Neumann algebra M , the presence of an A-B sub-bimodule of L(M) with finite right B-dimension implies that a ‘corner’ of A can be conjugated into B by a partial isometry in M . The technique was actually shown by Popa to apply to the case of (M,φ) not necessarily finite (but with discrete decomposition), though one needs the additional assumption that A and B are subalgebras of the centralizer M. In this paper we will demonstrate how the above results can be extended to the case of free products of semiexact II1 factors with respect to arbitrary (non-tracial) states. The primary difficulty this setting presents is that the resulting factors will be necessarily of type III. Following the same basic outline of the proof from [Oz], we will therefore utilize a generalization of Popa’s intertwining technique that allows the ambient algebra M to be completely arbitrary, and makes no assumption about the relative position of the subalgebras A and B in M . One of the fruits of our labors will be the result that for Mi semiexact non-prime non-injective II1 factors with faithful normal states φi, the reduced free product ∗mi=1(Mi, φi) of the (Mi, φi) is uniquely written in such a way as the free product of such factors, up to permutation of indices and stable isomorphism. Acknowledgement: The author would like to thank Dimitri Shlyakhtenko for initially suggesting this investigation, and for his invaluable subsequent advice and direction.