Conditional expectations on Fell bundles

TL;DR

This paper demonstrates that continuous conditional expectations between Fell bundles induce embeddings and conditional expectations between their cross-sectional C*-algebras, also relating amenability properties.

Contribution

It establishes the implications of conditional expectations on Fell bundles for their associated C*-algebras and amenability, extending known results.

Findings

Full cross-sectional C*-algebra containment under conditional expectation

Existence of conditional expectations between cross-sectional C*-algebras

Amenability of subbundles follows from bundle amenability

Abstract

We show that the existence of a continuous conditional expectation from a Fell bundle to a Fell subbundle implies that the full cross-sectional C*-algebra of the subbundle is contained in the full cross-sectional C*-algebra of the bundle, and moreover there exists a conditional expectation from the latter algebra to the former. A similar fact holds for the reduced cross-sectional C*-algebras. Besides, the subbundle is amenable whenever so is the bundle.