Unitary representations of the Roe algebra of a discrete group and symmetries

TL;DR

This paper studies unitary representations of Roe algebras associated with discrete groups, establishing conditions under which these representations factor through reduced group C*-algebras, and deriving implications for the structure of certain group von Neumann algebras.

Contribution

It introduces new conditions on group actions that ensure representations of Roe algebras factor through reduced group C*-algebras, with applications to the classification of von Neumann algebras.

Findings

Groups like SL_3(Z) and PGL_2(Z[1/p]) have the Akemann-Ostrand property.

Certain group von Neumann algebras are proven to be non-isomorphic.

The paper establishes factorization of representations under specific dynamical and topological conditions.

Abstract

Let $\Gamma$ be a discrete countable group. Consider the crossed product C$^\ast$-algebra $\mathfrak{R}(\Gamma) = C^{\ast}(\Gamma \rtimes l^{\infty}(\Gamma))$. Let $G$ be a larger discrete group, containing $\Gamma$ as an almost normal subgroup. Consequently $G$ acts by partial isomorphisms on $G$ and hence on $\mathfrak {R}(\Gamma)$. Let $\mathfrak{R}_G(\Gamma)$ be the crossed product $C^{\ast}$ - algebra $C^{\ast}(G \times (\mathfrak{R}(\Gamma))$. The C$^\ast$-algebra $\mathfrak{R}_G(\Gamma)$ has a natural representation into $\mathcal B(\ell ^2(\Gamma))$ and hence also admits a representation $\Pi_{\mathcal{Q}}$ into the Calkin algebra $\mathcal{Q}(\ell ^2(\Gamma))$. Let $G\rtimes \Gamma=\Gamma\times \Gamma^{\rm op} $ and assume that $\Gamma$ is exact. Assume that the non-trivial conjugation orbits under the action of $\Gamma$, having non amenable stabilizers, are separated, in a suitable chosen profinite topology, from the identity element in $\Gamma$. We also assume natural amenability conditions on the dynamics of the action of $\Gamma\times \Gamma^{\rm op}$ on cosets of amenable subgroups. Then $\Pi_{\mathcal Q}$ factorises to a representation of $C^{\ast}_{\rm red}(G \rtimes \mathfrak{R}(\Gamma))$. In particular the groups ${\mathop{SL}}_3(\mathbb Z)$, ${\mathop{\rm PGL}}_2(\mathbb Z[\frac{1}{p}])$ have the Akemann-Ostrand property. This implies, using the solidity property of Ozawa ([Oz]), that the group von Neumann algebras, $\mathcal L({\mathop{SL}}_3(\mathbb Z))$ and $\mathcal L({\mathop{SL}}_n(\mathbb Z))$, $n\geq 4$, are non-isomorphic.