Induction and absorption of representations and amenability of Banach *-algebraic.

Given a Fell bundle $\mathcal{B}$ (saturated or not) over $G$ and a closed subgroup $H\subset G,$ we prove that any *-representation of the reduction $\mathcal{B}_H$ can be induced to $\mathcal{B}.$ We observe that Exel-Ng's reduced cross sectional C*-algebra $C^*_r(\mathcal{B})$ is universal for the *-representations induced from $B_e=\mathcal{B}_{\{e\}}$ and construct a cross sectional C*-algebra of $\mathcal{B},$ $C^*_H(\mathcal{B}),$ that is universal for the *-representations induced from $\mathcal{B}_H.$ We prove an absorption principle for $C^*_H(\mathcal{B})$ with respect to tensor products of *-representations of $\mathcal{B}$ and *-representations of $G$ induced from $H.$ Using this principle we show, among other results, that given closed normal subgroups of $G,$ $H\subset K,$ there exists a quotient map $q^{\mathcal{B}}_{KH}\colon C^*_K(\mathcal{B})\to C^*_H(\mathcal{B})$ which is a C*-isomorphism if and only if $q^{\mathcal{B}_K}_{KH}\colon C^*(\mathcal{B}_K)\to C^*_H(\mathcal{B}_K)$ is a C*-isomorphism. We also prove the two conditions above hold if $q^G_{KH}\colon C^*_K(G)\to C^*_H(G)$ is a C*-isomorphism. All the constructions are performed using Banach *-algebraic bundles having a strong approximate unit.