The conjugacy relation on unitary representations

Abstract

Abstract. In a 1965 paper, E.G. Effros asked the question if the conjugacy relation for unitary representations of a locally compact second countable group is a Borel equivalence relation. In this paper, we answer this question affirmatively. This also settles a recent question raised by A.S. Kechris regarding the hierarchy complexity of unitary equivalence of probability measure preserving actions of countable discrete groups.

The Borel complexity of von Neumann equivalence

“…The hypothesis on Γ can be weakened considerably, and we will state our result in full in the next section. We note that Theorem 1.2 and the result of Foreman, Rudolph and Weiss stand in contrast to the recent result of Hjorth and Törnquist [16], where it is shown that conjugacy of unitary representations of any countably infinite discrete Γ is always Borel.…”

Set theory and von Neumann algebras

“…The details can be found in [HT12]. The theorem applies more generally to representations of locally compact second countable groups and separable C*-algebras.…”