Renault's $j$-map for Fell bundle $C^*$-algebras

Abstract

If $p \colon \mathcal B\to G$ is a Fell bundle over an \'etale groupoid, then we show that there is an norm reducing injective linear map $j \colon C^*_r(G;\mathcal B)\to \Gamma_{0}(G;\mathcal B)$ generalizing the well know map $j \colon C^*_{r}(G)\to C_{0}(G)$ in the case of an \'etale groupoid.