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

TL;DR

This paper introduces a new injective linear map called the $j$-map for Fell bundle $C^*$-algebras over étale groupoids, generalizing a classical map in the theory of groupoid $C^*$-algebras.

Contribution

It constructs a norm reducing injective linear map from reduced Fell bundle $C^*$-algebras to sections, extending known results for groupoid $C^*$-algebras.

Findings

The $j$-map is norm reducing and injective.

Generalizes the classical $j$-map for groupoid $C^*$-algebras.

Provides a new tool for analyzing 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.