Fell algebras, groupoids, and projections

TL;DR

This paper constructs examples of Fell algebras with specific properties to explore differences from continuous trace $C^*$-algebras, focusing on the existence of full projections and spectrum conditions.

Contribution

It introduces new examples of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant, highlighting their distinct features from continuous trace algebras.

Findings

Fell algebras with locally Hausdorff spectrum are constructed.

The existence of full projections in these algebras is analyzed.

Applications to dynamical systems and Wieler solenoids are discussed.

Abstract

Examples of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant are constructed to illustrate differences with the case of continuous trace $C^*$-algebras. At the level of the spectrum, this translates to only assuming the spectrum is locally Hausdorff (rather than Hausdorff). The existence of (full) projections is the fundamental question considered. The class of Fell algebras studied here arise naturally in the study of Wieler solenoids and applications to dynamical systems will be discussed in a separate paper.