Sections of Fell Bundles over Étale Groupoids

We construct the reduced and essential C*-algebra of a Fell bundle over an étale groupoid (in full generality, without any second countability, local compactness or Hausdorff assumptions, even on the unit space) directly from sections under convolution. This eliminates the j-map commonly seen in the groupoid and Fell bundle C*-algebra literature. We further show how multiplier algebras and other Banach -bimodules can again be constructed directly from sections. Finally, we extend Varela’s morphisms from C-bundles to Fell bundles, thus making the reduced C*-algebra construction functorial.