Product between ultrafilters and applications to Connes' embedding problem (Q2919620)

Let \(\mathcal U,\mathcal V\) be two ultrafilters on \(I,J\), respectively. The tensor product \(\mathcal U\otimes\mathcal V\) is the ultrafilter on \(I\times J\) defined by NEWLINE\[NEWLINEX\in\mathcal U\otimes\mathcal V\Leftrightarrow\{i\in I:\{j\in J: (i,j)\in X\}\in\mathcal V\}\in\mathcal U.NEWLINE\]NEWLINE The authors use this notion to investigate Connes' embedding problem. They 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 the ultrapower \(R^\omega\) of the hyperfinite type \({\text{II}}_1\) factor \(R\) with respect to a free ultrafilter \(\omega\) on \(\mathbb N\). They also show that the crossed product of a hyperlinear group via a profinite action is embeddable into \(R^\omega\).