Stable equivalence of the weak closures of free groups convolution algebras

Abstract

Abstract. We prove in this paper that the von Neumann algebras associated to the free non-commutative groups are stably isomorphic, i.e. that they are isomorphic when tensorized by the algebra of all linear bounded operators on a separable, infinite dimensional Hubert space. This gives positive evidence for an old question, due to R

The analogues of entropy and of Fisher's information measure in free probability theory, I

“…Thus, large random matrices provide an asymptotic model for free probability theory. This has had important applications to the study of the II X factors of free groups [30,19,8,20].…”

Free-independent sequences in type $II_1$ factors and related problems

“…For each Q = L(¥x), as defined in [Ra2], [Dy] for x £ (l,oo], and for each s £ {4cos2 7r/n\n > 4}, the factors MS(Q) were proved to be of the form £(Fy), for some у £ (l,oo] depending on x and s, in [Ra2]. But for s > 4 and/or Q = L(Vi) = L°°(T,//), Q = R, the problem of identifying MS(Q) (or its generalizations [Ba], [Ra3]) as some L(¥y) is still open. If answered positively, this would prove the existence of irreducible subfactors of any index s > 4 in L(¥x), 1 < x < oo.…”