Ideals in cross sectional $C^*$-algebras of Fell bundles

Abstract

Abstract. With each Fell bundle over a discrete group G we associate a partial action of G on the spectrum of the unit fiber. We discuss the ideal structure of the corresponding full and reduced cross-sectional C * -algebras in terms of the dynamics of this partial action.

Morita Enveloping Fell Bundles

“…Then there exists an isometry V : F ⊗ πA K → E ⊗ π H such that V (x⊗ πA k) = x⊗ π k for all x ∈ F , k ∈ K, as claimed in (1). Now, if a ′ ∈ A ′ , x ∈ F and k ∈ K:…”

“…Only discrete groups are considered in [1]. But in this section we prove that the partial actionα B is always continuous if G is a locally compact group.…”

“…We then show that the enveloping action of the spectral partial action associated to a Fell bundle A is precisely the global action on the spectrum of (A) induced by its canonical action α. In particular, many results of [1] can be obtained from the already existing results for crossed products by ordinary actions. In the same spirit we extend some results about amenability and nuclearity of crossed products to the realm of Fell bundles.…”

Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles

“…The partial homeomorphism of A associated to the identity Hilbert A-bimodule A is the identity on A, and the partial homeomorphism associated to X ⊗ B Y for two composable Hilbert bimodules is the product X • Y of partial homeomorphisms. Hence X * is the partial inverse X −1 of X (see [1,33]). …”

“…Aperiodic and topologically free Fell bundles over discrete groups have been defined already in [1,32,33]. Several other non-triviality conditions for group actions generalise readily to Fell bundles (see Definition 4.5 below).…”