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

Abstract

Let $\Gamma$ be a discrete countable group. Consider the crossed product C${}^{\ast }$-algebra $\mathfrak{R}(\Gamma )=C^{\ast }(\Gamma ⋊l^{\infty }(\Gamma ))$. 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 $G⋊\Gamma =\Gamma \times {\Gamma }^{\mathrm{op}}$ 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 $\Gamma \times {\Gamma }^{\mathrm{op}}$ on cosets of amenable subgroups. Then ${\Pi }_{\mathcal{Q}}$ factorises to a representation of $C_{\mathrm{red}}^{\ast }(G⋊\mathfrak{R}(\Gamma ))$. In particular the groups ${\mathrm{SL}}_3(\mathbb{Z})$, ${\mathrm{PGL}}_2(\mathbb{Z}[\frac{1}{p}])$ have the Akemann-Ostrand property. This implies, using the solidity property of Ozawa ([Oz]), that the group von Neumann algebras, $\mathcal{L}({\mathrm{SL}}_3(\mathbb{Z}))$ and $\mathcal{L}({\mathrm{SL}}_n(\mathbb{Z}))$, $n\ge 4$, are non-isomorphic.

arXiv comment40 pages, The paper has been entirely revised. The argument of proof were much simplified, by using the formalism of Loeb measures and non-standard universe to represent crossed product C* states in the Calkin algebra

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