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

TL;DR

This paper develops Fourier and Fourier-Stieltjes algebras for Fell bundles over discrete groups, establishing their properties, duality, and connections to amenability and $C^*$-dynamical systems.

Contribution

It introduces and analyzes the Fourier and Fourier-Stieltjes spaces for Fell bundles, extending classical harmonic analysis concepts to this setting with new structural results.

Findings

$B(B)$ is isomorphic to the dual of $C^*(B)$ for saturated bundles.

$A(B)$ and $B(B)$ are Banach algebras under certain conditions.

Amenability of $G$ implies bounded approximate identity for $A(B)$.

Abstract

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 $C^{*}$-algebra $C^{*}(\mathcal{B})$ 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 $C^*$-algebra or $\mathcal{B}$ is the Fell bundle of a $C^*$-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 $C^*$-dynamical system $(A,G,\gamma)$. The converse is left as an open problem.