A descriptive view of unitary group representations (Q496464)

The main result of the paper is Theorem 1.11: If \(G\) is a countable non-type \(I\) group and \(H\) is a countable amenable group, then \(\thickapprox_H\) is Borel reducible to \(\thickapprox_G\). If \(G\) and \(H\) are countable amenable non-type \(I\) groups, then \(\thickapprox_G\) and \(\thickapprox_H\) are Borel bireducible. It is proved that the universal countable Borel equivalence relation \(E_{\infty}\) is Borel reducible to the unitary equivalence relation \(\thickapprox_{\mathbb{F}_2}\) on the two generators \(\{a,b\}\). In the final section the unitary relation \(\thickapprox^+ _G\) is considered on the space \(\mathrm{Rep}(G,\mathcal{H})\) of arbitrary unitary representations of \(G\) on \(\mathcal{H}\).