The von Neumann algebra of the non-residually finite Baumslag group < a,b | a b^3 a^-1 = b^2 > embeds into R^omega

Abstract

In this paper we analyze the structure of some sets of non-commutative moments of elements in a finite von Neumann algebra M. If the fundamental group of M is R_+\{0}, then the moment sets are convex, and if M is isomorphic to M tensor M, then the sets are closed under pointwise multiplication. We introduce a class of discrete groups that we call hyperlinear. These are the discrete subgroups (with infinite conjugacy classes) of the unitary group of R^omega. We prove that this class is strictly larger than the class of (i.c.c.) residually finite groups. In particular, it contains the Baumslag group < a,b | a b^3 a^-1 = b^2 >. This leads to a previously unknown (non-hyperfinite) type II_1 factor that can be embedded in R^omega. This is positive evidence for Connes's conjecture that any separable II_1 factor can be embedded into R^omega.