Fell algebras, groupoids, and projections

Examples of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant are constructed to illustrate differences with the case of continuous trace $C^{\ast }$-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.