Remarks on the Ideal Structure of Fell Bundle C*-Algebras

TL;DR

This paper investigates the structure of Fell bundle C*-algebras over groupoids, establishing a continuous groupoid action on primitive ideal spaces and deriving exact sequences for invariant ideals, generalizing known results from dynamical systems.

Contribution

It introduces a continuous groupoid action on the primitive ideal space of Fell bundle C*-algebras and derives exact sequences for invariant ideals, extending classical dynamical system results.

Findings

Existence of a continuous G-action on Prim A

Short exact sequences for G-invariant ideals in Fell bundle C*-algebras

Generalization of classical results from dynamical systems

Abstract

We show that if $p:\B\to G$ is a Fell bundle over a locally compact groupoid $G$ and that $A=\Gamma_{0}(G^{(0)};\B)$ is the \cs-algebra sitting over $G^{(0)}$, then there is a continuous $G$-action on $\Prim A$ that reduces to the usual action when $\B$ comes from a dynamical system. As an application, we show that if $I$ is a $G$-invariant ideal in $A$, then there is a short exact sequence of \cs-algebras \xymatrix{0\ar[r]&\cs(G,\BI)\ar[r] &\cs(G,\B)\ar[r]&\cs(G,\BqI)\ar[r]&0,} where $\cs(G,\B)$ is the Fell bundle \cs-algebra and $\BI$ and $\BqI$ are naturally defined Fell bundles corresponding to $I$ and $A/I$, respectively. Of course this exact sequence reduces to the usual one for \cs-dynamical systems.