Covariant representations for possibly singular actions on C*-algebras

TL;DR

This paper surveys and consolidates existing results on covariant representation theory for singular group actions on C*-algebras, extending classical concepts to non-locally compact and non-strongly continuous cases.

Contribution

It compiles, clarifies, and improves proofs of key results in the covariant representation theory for singular C*-actions, filling gaps in the literature.

Findings

Includes existence theorems by Borchers and Halpern

Analyzes Arveson spectra and the Borchers-Arveson theorem

Examines ground states, KMS states, and their GNS representations

Abstract

Singular actions on C*-algebras are automorphic group actions on C*-algebras, where the group need not be locally compact, or the action need not be strongly continuous. We study the covariant representation theory of such actions. In the usual case of strongly continuous actions of locally compact groups on C*-algebras, this is done via crossed products, but this approach is not available for singular C*-actions (this was our path in a previous paper). The literature regarding covariant representations for singular actions is already large and scattered, and in need of some consolidation. We collect in this survey a range of results in this field, mostly known. We improve some proofs and elucidate some interconnections. These include existence theorems by Borchers and Halpern, Arveson spectra, the Borchers-Arveson theorem, standard representations and Stinespring dilations as well as ground states, KMS states and ergodic states and the spatial structure of their GNS representations.