Duality for full crossed products of 𝐶*-algebras by non-amenable groups

Abstract

Abstract. Let (A, G, δ) be a cosystem and (A, G, α) be a dynamical system. We examine the extent to which induction and restriction of ideals commute, generalizing some of the results of Gootman and Lazar (1989) to full crossed products by non-amenable groups. We obtain short, new proofs of Katayama and Imai-Takai duality, the faithfulness of the induced regular representation for full coactions and actions by amenable groups. We also give a short proof that the space of dual-invariant ideals in the crossed product is homeomorphic to the space of invariant ideals in the algebra, and give conditions under which the restriction mapping is open.

Approximation Properties for Crossed Products by Actions and Coactions

“…This follows from the work of Raeburn, [27], who showed that the full crossed product, defined by universal properties, is isomorphic to the reduced crossed product that we have defined here. The papers [7,18,20,22,27] are good references for background material.…”

Stone duality and quasi‐orbit spaces for generalised C∗‐inclusions

“…Hence, this assumption is necessary for our proof to work. That restricted to coincides with is proved in [26, Proposition 3.14(iii)] if amenable (see also [47, Proposition 3.1(iii)], where the full crossed products are considered).…”

“…Remark It is known that a diagram similar to () commutes when is replaced by the map given by kernels of the corresponding induced representations (see [26, Remarks 2.8] or [47, Propositions 2.7 and 2.8]). Thus, we need to show that and coincide on restricted ideals.…”

Crossed Products of Operator Algebras

“…Remark 3.24. Certainly the extension of [39, Corollary 2.2] to non-separable systems does not require the heavy machinery of coactions, as it happens in [76,Corollary 3.4 (i)]. Nevertheless we have not able to locate an appropriate reference in the literature that does not involve coactions.…”