Covariant representations for possibly singular actions on $C^*$-algebras
Singular actions on $C^*$-algebras are automorphic group actions on $C^*$-algebras, where the group is not locally compact, or the action is not strongly continuous. We study the covariant representation theory of actions which may be singular. 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. We explored extension of crossed products to singular actions in a previous paper. The literature regarding covariant representations for possibly 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.