Crossed products of C*-algebras for singular actions

TL;DR

This paper develops a generalized framework called 'crossed product host' for analyzing singular group actions on C*-algebras, extending the classical theory to non-strongly continuous and non-locally compact group actions, with applications in physics models.

Contribution

It introduces the concept of crossed product hosts for singular actions, establishing their uniqueness, existence conditions, and providing examples where they differ from traditional crossed products.

Findings

Crossed product hosts exist even when traditional crossed products do not.

A uniqueness theorem for crossed product hosts is proven.

The framework applies to discontinuous actions of locally compact groups.

Abstract

We consider group actions of topological groups on C*-algebras of the types which occur in many physics models. These are singular actions in the sense that they need not be strongly continuous, or the group need not be locally compact. We develop a "crossed product host" in analogy to the usual crossed product for strongly continuous actions of locally compact groups, in the sense that its representation theory is in a natural bijection with the covariant representation theory of the action. We prove a uniqueness theorem for crossed product hosts, and analyze existence conditions. We also present a number of examples where a crossed product host exists, but the usual crossed product does not. For actions where a crossed product host does not exist, we obtain a "maximal" invariant subalgebra for which a crossed product host exists. We further study the case of a discontinuous action of a locally compact group in detail.