Cross-sectional C*-algebras associated to subgroups

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 $C^*_{H\uparrow \mathcal{B}}(\mathcal{B})$ and $C^*_{H\uparrow G}(\mathcal{B})$ of the full cross-sectional C*-algebra $C^*(\mathcal{B})$ analogous to Exel-Ng's reduced algebras $C^*_{\mathop{\rm r}}(\mathcal{B})\equiv C^*_{\{e\}\uparrow \mathcal{B}}(\mathcal{B})$ and $C^*_R(\mathcal{B})\equiv C^*_{\{e\}\uparrow G}(\mathcal{B}).$ 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 $C^*_{H\uparrow \mathcal{B}}(\mathcal{B})=C^*_{H\uparrow G}(\mathcal{B}).$ 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 $C^*(\mathcal{B}_H)=C^*_{\mathop{\rm r}}(\mathcal{B}_H)$ if and only if $C^*_{H\uparrow \mathcal{B}}(\mathcal{B})=C^*_{\mathop{\rm r}}(\mathcal{B});$ the last identification being implied by $C^*(\mathcal{B})=C^*_{\mathop{\rm r}}(\mathcal{B}).$ We also prove that if $G$ is inner amenable and $C^*_{\mathop{\rm r}}(\mathcal{B})\otimes_{\max} C^*_{\mathop{\rm r}}(G)=C^*_{\mathop{\rm r}}(\mathcal{B})\otimes C^*_{\mathop{\rm r}}(G),$ then $C^*(\mathcal{B})=C^*_{\mathop{\rm r}}(\mathcal{B}).$