Fourier and Fourier-Stieltjes algebra of Fell bundles over discrete groups

For a Fell bundle $\mathcal{B}={\left\{B_s\right\}}_{s\in G}$ over a discrete group $G$, we use representations theory of $\mathcal{B}$ to construct the Fourier and Fourier-Stieltjes spaces $A(\mathcal{B})$ and $B(\mathcal{B})$ of $\mathcal{B}$. When $\mathcal{B}$ is saturated we show $B(\mathcal{B})$ is canonically isomorphic to the dual space of the cross sectional $C^{\ast }$-algebra $C^{\ast }(\mathcal{B})$ of $\mathcal{B}$. When there is a compatible family of co-multiplications on the fibers we show that $B(\mathcal{B})$ and $A(\mathcal{B})$ are Banach algebras. This holds in particular if either the fiber $B_e$ at identity is a Hopf $C^{\ast }$-algebra or $\mathcal{B}$ is the Fell bundle of a $C^{\ast }$-dynamical system. When $A(\mathcal{B})$ is a Banach algebra with bounded approximate identity, we show that $B(\mathcal{B})$ is the multiplier algebra of $A(\mathcal{B})$. We prove a Leptin type theorem by showing that amenability of $G$ implies the existence of bounded approximate identity for $A(\mathcal{B})$ for bundles coming from a $C^{\ast }$-dynamical system $(A,G,\gamma )$. The converse is left as an open problem.