Cross-sectional C*-algebras associated with subgroups

Abstract Given a Fell bundle $\mathcal {B}=\{B_t\}_{t\in G}$ over a locally compact 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 $C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B})\equiv C^{*}_{\{e\}\uparrow \mathcal {B}}(\mathcal {B})$ 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 $C^{*}(\mathcal {B}_H)=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B}_H)$ if and only if $C^{*}_{H\uparrow \mathcal {B}}(\mathcal {B})=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B});$ the last identification being implied by $C^{*}(\mathcal {B})=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B}).$ We also prove that if G is inner amenable and $C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B})\otimes _{\max } C^{*}_{\mathop {\mathrm {r}}}(G)=C^{*}_{\mathop {\mathrm { r}}}(\mathcal {B})\otimes C^{*}_{\mathop {\mathrm {r}}}(G),$ then $C^{*}(\mathcal {B})=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B}).$