Topological freeness for Hilbert bimodules

Abstract

Abstract. It is shown that topological freeness of Rieffel's induced representation functor implies that any C * -algebra generated by a faithful covariant representation of a Hilbert bimodule X over a C * -algebra A is canonically isomorphic to the crossed product A ⋊ X Z. An ideal lattice description and a simplicity criterion for A ⋊ X Z are established. IntroductionThe topological freeness is a condition expressed in terms of the dual system allowing to relate the ideal structure of the crossed product to that of the original algebra. In particular, it implies that every faithful representation of the C * -dynamical system integrates to a faithful representation of the reduced crossed product. The idea behind this notion probably goes back to works of W. Arveson in late 60's of XX century, and for the first time was explicitly formulated by D. [12, Thm. 3.7].In the present paper we prove a statement that generalizes all the aforementioned theorems in the case G = Z and which is formulated in terms of the crossed product A ⋊ X Z, introduced in [1], of a Hilbert bimodule X. Thus potentially, by passing to the core C * -algebra, see [1, Thm. 3.1], it may be applied to all the C * -algebras equipped with a semi-saturated circle action and thereby to all relative CuntzPimsner algebras [13]. As a corollary of our main result we provide an ideal lattice description (in the case the dual system is free) and a simplicity criterion for the algebras considered.Conventions. In essence we follow the notation and conventions adopted in [1]. For maps γ : A × B → C such as inner products, multiplications or representations we denote by γ(A, B) the closed linear span of the set {γ(a, b) ∈ C : a ∈ A, b ∈ B}. An ideal in a C * -algebra is a closed two-sided one, and [π] stands for the unitary equivalence class of a representation π.2010 Mathematics Subject Classification. Primary 46L08, Secondary 46L55.

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

“…Archbold and Spielberg discussed in [8] the relation between the ideal structure of the full crossed product and that of the base algebra, under the assumption of topological freeness. More recently, the definition of topological freeness and several related results were extended to different settings: by Exel, Laca and Quigg for partial actions on commutative C * -algebras in [12], by Lebedev in [17], and later by Giordano and Sierakowski in [14], for partial actions on arbitrary C * -algebras, and by Kwaśniewski in [16]) for crossed products by Hilbert C * -bimodules.…”

“…After establishing some background and notation in Section 1, we introduce in Section 2 a partial actionα on the spectrum of the unit fiber of a Fell bundle B over a discrete group. When B is the Fell bundle corresponding to a partial action γ, thenα agrees withγ, as defined in [5,Section 7] or [17], and when B is the Fell bundle associated in [2] with the crossed-product by a Hilbert C * -bimodule, thenα is the homeomorphismĥ discussed in [16]. Following familiar lines, we establish in Section 3 a bijective correspondence between the family ofα-invariant open sets in the spectrum of the unit fiber and the set of ideals in B (Proposition 3.8 and Proposition 3.10) This enables us to show that, whenα is topologically free, its minimality is equivalent to the simplicity of C * r (B) (Corollary 3.12).…”

“…Corollary 3. 16. If the partial action of the Fell bundle B is topologically free on every invariant closed subset ofB e , then B has the residual intersection property.…”

Crossed products by endomorphisms of C0(X)-algebras

“…For reversible and extendible systems the relevant statements in [33,Subsection 4.5] were deduced from [34, Theorem 2.20]. We will extend them by applying general results from [32] and facts presented in Appendix A.…”

“…where α´1 x is the inverse to the isomorphism α x : pker α x q K Ñ α x p1 ϕpxq qApxqα x p1 ϕpxq q. The maps α˚, x have strictly continuous extensions which are given by (32) with α´1 x replaced by the inverse to the strictly continuous isomorphism α x : Mppker α x q K q Ñ α x p1 ϕpxq qMpApxqqα x p1 ϕpxq q, cf. Lemma 3.12.…”