The Hecke algebra of a Hecke pair (G, H) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C*-completions of are addressed in terms of the projection using both Fell's and Rieffel's imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.

It is shown that if G is a compact ergodic group of *-automorphisms on a unital c*-algebra A then the un1que G-invariant state is a trace. Hence if A is a von Neumann algebra then it is finite.

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.

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If (A, p, G) is a covariant system over a locally compact group G, i.e. p is a homomorphism from G into the group of '-automorphisms of an operator algebra A, there is a new operator algebra 21 called the covariance algebra associated with (A, p, G). If A is a von Neumann algebra and p is a-weakly continuous, Sf is defined such that it is a von Neumann algebra. If A is a C*-algebra and p is norm-continuous St will be a C*-algebra. The following problems are studied in these two different settings: 1. If 31 is a covariance algebra, how do we recover A and p? 2. When is an operator algebra St the covariance algebra for some covariant system over a given locally compact group G?