Cross-sectional C*-algebras associated to subgroups

Abstract

Given a Fell bundle $\mathcal{B}=\{B_t{\}}_{t\in G}$ over a locally compact and Hausdorff group $G$ and a closed subgroup $H\subset G,$ we construct quotients $C_{H\uparrow \mathcal{B}}^{\ast }(\mathcal{B})$ and $C_{H\uparrow G}^{\ast }(\mathcal{B})$ of the full cross-sectional C*-algebra $C^{\ast }(\mathcal{B})$ analogous to Exel-Ng's reduced algebras $C_r^{\ast }(\mathcal{B})\equiv C_{\{e\}\uparrow \mathcal{B}}^{\ast }(\mathcal{B})$ and $C_R^{\ast }(\mathcal{B})\equiv C_{\{e\}\uparrow G}^{\ast }(\mathcal{B}).$ An absorption principle, similar to Fell's one, is used to give conditions on $\mathcal{B}$ and $H$ (e.g. $G$ discrete and $\mathcal{B}$ saturated, or $H$ normal) ensuring $C_{H\uparrow \mathcal{B}}^{\ast }(\mathcal{B})=C_{H\uparrow G}^{\ast }(\mathcal{B}).$ The tools developed here enable us to show that if the normalizer of $H$ is open in $G$ and ${\mathcal{B}}_H:=\{B_t{\}}_{t\in H}$ is the reduction of $\mathcal{B}$ to $H,$ then $C^{\ast }({\mathcal{B}}_H)=C_r^{\ast }({\mathcal{B}}_H)$ if and only if $C_{H\uparrow \mathcal{B}}^{\ast }(\mathcal{B})=C_r^{\ast }(\mathcal{B});$ the last identification being implied by $C^{\ast }(\mathcal{B})=C_r^{\ast }(\mathcal{B}).$ We also prove that if $G$ is inner amenable and $C_r^{\ast }(\mathcal{B}){\otimes }_{\max }{C}_r^{\ast }(G)=C_r^{\ast }(\mathcal{B})\otimes C_r^{\ast }(G),$ then $C^{\ast }(\mathcal{B})=C_r^{\ast }(\mathcal{B}).$