Journal of Operator Theory

Volume 68, Issue 1, Summer 2012 pp. 165-172.

Product between ultrafilters and applications to Connes' embedding problem

Authors: Valerio Capraro (1) and Liviu Paunescu (2)
Author institution: (1) Dipartimento di Matematica, Universita degli Studi di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy
(2) Dipartimento di Matematica, Universita degli Studi di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy and Institute of Mathematics "S. Stoilow" of the Romanian Academy, PO Box 1-764, 014700 Bucharest, Romania

Summary: In this paper we want to apply the notion of product between ultrafilters to answer several questions which arise around Connes' embedding problem. We shall prove that an ultraproduct of hyperlinear groups is still hyperlinear and consequently the von Neumann algebra of the free group with uncountable many generators is embeddable into $R^{\omega}$. This follows also from a general construction that allows, starting from an hyperlinear group, to find a family of hyperlinear groups. We shall prove also that the cross product of an hyperlinear group via a profinite action is embeddable into $R^{\omega}$.

Keywords: Hyperlinear groups, Connes' embedding problem, product of ultrafilters

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