Product between ultrafilters and applications to the Connes' embedding problem

In this paper we want to apply the notion of product between ultrafilters to answer several questions which arise around the Connes' embedding problem. For instance, we will give a simplification and generalization of a theorem by Radulescu; we will prove that 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 will introduce the notion of hyperlinear pair and we will use it to give some other characterizations of hyperlinearity. We will prove also that the cross product of a hyperlinear group via a profinite action is embeddable into $R^{\omega}$.