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

Abstract

Let $Γ$ be a discrete countable group. Consider the crossed product C$^\ast$-algebra $\mathfrak{R}(Γ) = C^{\ast}(Γ\rtimes l^{\infty}(Γ))$. Let $G$ be a larger discrete group, containing $Γ$ as an almost normal subgroup. Consequently $G$ acts by partial isomorphisms on $G$ and hence on $\mathfrak {R}(Γ)$. Let $\mathfrak{R}_G(Γ)$ be the crossed product $C^{\ast}$ - algebra $C^{\ast}(G \times (\mathfrak{R}(Γ))$. The C$^\ast$-algebra $\mathfrak{R}_G(Γ)$ has a natural representation into $\mathcal B(\ell ^2(Γ))$ and hence also admits a representation $Π_{\mathcal{Q}}$ into the Calkin algebra $\mathcal{Q}(\ell ^2(Γ))$. Let $G\rtimes Γ=Γ\times Γ^{\rm op} $ and assume that $Γ$ is exact. Assume that the non-trivial conjugation orbits under the action of $Γ$, having non amenable stabilizers, are separated, in a suitable chosen profinite topology, from the identity element in $Γ$. We also assume natural amenability conditions on the dynamics of the action of $Γ\times Γ^{\rm op}$ on cosets of amenable subgroups. Then $Π_{\mathcal Q}$ factorises to a representation of $C^{\ast}_{\rm red}(G \rtimes \mathfrak{R}(Γ))$. 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.