R-coactions on $C^*$-algebras

TL;DR

This paper introduces the concept of R-coactions on $C^*$-algebras, exploring their foundational properties and potential applications in the context of the Baum-Connes conjecture.

Contribution

It develops the initial theory of R-coactions, highlighting differences from standard coactions and establishing basic properties for future research.

Findings

Identified gaps in the R-coactions theory compared to standard coactions.

Developed foundational properties of R-coactions.

Outlined potential applications in Baum-Connes conjecture studies.

Abstract

We give the beginnings of the development of a theory of what we call "R-coactions" of a locally compact group on a $C^*$-algebra. These are the coactions taking values in the maximal tensor product, as originally proposed by Raeburn. We show that the theory has some gaps as compared to the more familiar theory of standard coactions. However, we indicate how we needed to develop some of the basic properties of R-coactions as a tool in our program involving the use of coaction functors in the study of the Baum-Connes conjecture.