  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
$C^*(\mathcal{B})=C^*_{\mathop{\rm r}}(\mathcal{B}).$
