Integrable and proper actions on C*-algebras, and square-integrable representations of groups

TL;DR

This paper defines proper group actions on C*-algebras, explores their properties, and connects these ideas to square-integrable group representations, extending classical notions to a noncommutative setting.

Contribution

It introduces a new definition of proper actions on C*-algebras, analyzes generalized fixed-point algebras, and links these concepts to square-integrable representations of groups.

Findings

The proposed definition captures classical proper actions in the commutative case.

A candidate for the generalized fixed-point algebra is identified, but shown to be too large in general.

Application to actions on compact operators provides insights into square-integrable representations.

Abstract

We propose a definition of what should be meant by a {\it proper} action of a locally compact group on a C*-algebra. We show that when the C*-algebra is commutative this definition exactly captures the usual notion of a proper action on a locally compact space. We then discuss how one might define a {\it generalized fixed-point algebra}. The goal is to show that the generalized fixed-point algebra is strongly Morita equivalent to an ideal in the crossed product algebra, as happens in the commutative case. We show that one candidate gives the desired algebra when the C*-algebra is commutative. But very recently Exel has shown that this candidate is too big in general. Finally, we consider in detail the application of these ideas to actions of a locally compact group on the algebra of compact operators (necessarily coming from unitary representations), and show that this gives an attractive view of the subject of square-integrable representations.