Fixed point algebras for weakly proper Fell bundles

TL;DR

This paper introduces weakly proper Fell bundles and develops multiple methods to construct exotic fixed point algebras, generalizing previous work and establishing Morita equivalences under certain conditions.

Contribution

It defines weakly proper Fell bundles, constructs exotic fixed point algebras, and generalizes existing theories on proper actions and their amenability.

Findings

Exotic fixed point algebras are constructed via three methods.

Under freeness, all exotic cross sectional C*-algebras are Morita equivalent to fixed point algebras.

Generalizes proper C*-actions to Fell bundles and proves their amenability.

Abstract

We define weakly proper Fell bundles and construct exotic fixed point algebras for such bundles. Three alternative constructions of such algebras are given. Under a kind of freeness condition, one of our constructions implies that every exotic cross sectional C*-algebra of a weakly proper Fell bundle is Morita equivalent to an exotic fixed point algebras. The other constructions are used to show that ours generalizes that of Buss and Echterhoff on weakly proper actions on C*-algebras. We also generalize to Fell bundles the fact that every C*-action which is proper in Kasparov's sense is amenable.