Representations of Crossed Products by Coactions and Principal Bundles

Abstract

Abstract. The main purpose of this paper is to establish a covariant representation theory for coactions of locally compact groups on C*-algebras (including a notion of exterior equivalence), to show how these results extend the usual notions for actions of groups on C*-algebras, and to apply these ideas to classes of coactions termed pointwise unitary and locally unitary to obtain a complete realization of the isomorphism theory of locally trivial principal G-bundles in this context. We are also able to obtain all (Cartan) principal G-bundles in this context, but their isomorphism theory remains elusive.

Applications of Non-Commutative Duality to Crossed Product C* -Algebras Determined by an Action or Coaction

“…For the convenience of the reader, we establish notation and summarize the main results from [8,10,19] which we shall be using. Let G be a locally compact group with modular function A.…”

“…Let 6 be a coaction of G on A. We employ in § 2 the representation theory of crossed coproduct algebras developed in [10] to define, given a representation K of A, an induced representation Ind n of G x 6 A, and, given a representation L of G x 6 A, the restriction Res" L of L to a representation of A. If G is abelian, so that the coaction of G determines an action a (8) of G, with Gx s A = G x a ( 6 ) i4, we obtain, of course, the usual representations Ind K and Res L [5,6].…”

Crossed products of C∗-correspondences by amenable group actions

“…Thus, ψ −1 (K(X γ G)) isα-invariant (by [15,Remarks 4.2(3)] and the amenability of G). Similarly, ker ψ areα-invariant.…”

“…(becauseα(b · w) = 0 for any w ∈ ker ψ ), and J X γ G isα-invariant (by [15,Remarks 4.2(3)] and the amenability of G).…”

“…[15]) and is called the dual coaction of α. As G is amenable,α is nondegenerate in the sense that the linear span of…”