Crossed Products by Dual Coactions of Groups and Homogeneous Spaces

Mansfield showed how to induce representations of crossed products of C∗algebras by coactions from crossed products by quotient groups and proved an imprimitivity theorem characterising these induced representations. We give an alternative construction of his bimodule in the case of dual coactions, based on the symmetric imprimitivity theorem of the third author; this provides a more workable way of inducing representations of crossed products of C∗-algebras by dual coactions. The construction works for homogeneous spaces as well as quotient groups, and we prove an imprimitivity theorem for these induced representations. Coactions of groups on C-algebras, and their crossed products, were introduced to make duality arguments available for the study of dynamical systems involving actions of nonabelian groups. For these to be effective, one needs to understand the representation theory of crossed products by coactions. The most powerful tool we have was provided by Mansfield [12]: he showed how to induce representations from crossed products by quotient groups, and proved an imprimitivity theorem which characterises these induced representations. Unfortunately, Mansfield’s construction is complicated and technical. The Hilbert bimodule with which he defines induced representations is difficult to manipulate, and one is tempted to seek other realisations of this bimodule and the induced representations. Here we show that, at least for the dual coactions arising in the study of ordinary dynamical systems, there is an alternative bimodule built along more conventional lines from spaces of continuous functions with values in C-algebras. This bimodule will be easier to work with, and will allow us to induce representations from quotient homogeneous spaces as well as quotient groups. The core of our construction is a special case of the symmetric imprimitivity theorem of [15]. Suppose α is an action of a locally compact group G on a C-algebra A. For each closed subgroupH ofG, there is a diagonal action α⊗τ ofG onA⊗C0(G/H): if we identify A⊗C0(G/H) with C0(G/H,A) in the usual way, then (α⊗ τ)t(f)(sH) = f(t sH). We show in §1 that there is a natural Morita equivalence between an iterated crossed product (C0(G,A) ×α⊗τ G) ×H and the imprimitivity algebra C0(G/H,A) ×α⊗τ G of Green [6]. If H is normal, this imprimitivity algebra can be identified with the crossed product (A×αG)×α̂|G/H by the restriction of the dual coaction, and the iterated crossed product with ((A×αG)×α̂G)×̂̂ α|H ; the existence of our Morita equivalence is therefore predicted by Mansfield’s imprimitivity theorem, although his construction gives no hint that the bimodule can be realised as a completion of Cc(G× G,A). In §2, we shall discuss these Date: 20 August, 1996. 1991 Mathematics Subject Classification. 46L55, secondary 22D25.