Renault's $j$-map for Fell bundle $C^{\ast }$-algebras

Abstract

If $p:\mathcal{B}\to G$ is a Fell bundle over an étale groupoid, then we show that there is an norm reducing injective linear map $j:C_r^{\ast }(G;\mathcal{B})\to {\Gamma }_0(G;\mathcal{B})$ generalizing the well know map $j:C_r^{\ast }(G)\to C_0(G)$ in the case of an étale groupoid.