Actions of rigid groups on UHF-algebras

TL;DR

This paper constructs a continuum of distinct strongly outer actions of property (T) groups on UHF-algebras, revealing complex classification properties and linking group cohomology to operator algebra actions.

Contribution

It introduces a method to assign to any second countable abelian pro-p group a strongly outer action of a property (T) group on a UHF-algebra, demonstrating the complexity of classifying such actions.

Findings

Existence of continuum many non-cocycle conjugate strongly outer actions

Cocycle conjugacy relations are complete analytic sets, not Borel

Construction of outer actions with prescribed cohomology on the hyperfinite II_1 factor

Abstract

Let $\Lambda$ be a countably infinite property (T) group, and let $D$ be UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of $\Lambda$ on $D$. The proof consists in assigning, to any second countable abelian pro-$p$ group $G$, a strongly outer action of $\Lambda$ on $D$ whose (weak) cocycle conjugacy class completely remembers the group $G$. The group $G$ is reconstructed from the action through its (weak) 1-cohomology set endowed with a canonical pairing function. Our construction also shows the following stronger statement: the relations of conjugacy, cocycle conjugacy, and weak cocycle conjugacy of strongly outer actions of $\Lambda$ on $D$ are complete analytic sets, and in particular not Borel. The same conclusions hold more generally when $\Lambda$ is only assumed to contain an infinite subgroup with relative property (T), and for actions on (not necessarily simple) separable, nuclear, UHF-absorbing, self-absorbing C*-algebras with at least one trace. Finally, we use the techniques of this paper to construct outer actions on $R$ with prescribed cohomology. Precisely, for every infinite property (T) group $\Lambda$, and for every countable abelian group $\Gamma$, we construct an outer action of $\Lambda$ on $R$ whose 1-cohomology is isomorphic to $\Gamma$.