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

Abstract

Let ( A , G , δ ) (A, G, \delta ) be a cosystem and ( A , G , α ) (A, G,\alpha ) 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.