Introduction

The aim of this paper is to bring into evidence the usefulness of considering not only actions of compact groups on operator algebras, but also the dual notion of coaction. A lot of what we shall say is contained in much greater detail in three series of papers, due to be published in the near future: four papers on ergodic actions, two on product type actions and two on equivariant K–theory. These all have their rather primitive origins in the three chapters of my thesis. The other main proponent of coactions of compact groups is Adrian Ocneanu, and we shall make frequent reference to his still unpublished work.

We now briefly summarise the contents of the rest of this paper. In Section II we recall the basic definitions of coactions of compact groups on von Neumann and C* algebras. We present two examples of C* algebras which arise perhaps unexpectedly as crossed products by coactions, and show how this observation can be used to explore their structure. The basic idea here is an old one: to use symmetry properties to simplify and elucidate computations. In Section III we exhibit two general principles in equivariant KK–theory, namely Frobenius Reciprocity and Dirac Induction. When combined with the equivariant Thorn isomorphism, these lead to a generalisation of a spectral theorem of Hodgkin (for the K–theory of spaces) which in principle provides a homological machine whereby ordinary KK-theory (of a pair of algebras) can be deduced from equivariant KK–theory.