Crossed products of C^*-algebras for singular actions with spectrum conditions

TL;DR

This paper investigates the existence and structure of crossed product constructions for singular Lie group actions on C*-algebras, including non-compact and non-strongly continuous actions, with applications to physics.

Contribution

It introduces the concept of cross representations for singular actions, analyzes their stability under perturbations, and extends the Borchers-Arveson theorem to non-abelian groups.

Findings

Existence of cross representations is stable under perturbations for one-parameter groups.

Cross property of higher-dimensional actions reduces to one-parameter subsystems.

Full extension of Borchers-Arveson theorem in the presence of cyclic invariant vectors.

Abstract

We analyze existence of crossed product constructions of Lie group actions on C^*-algebras which are singular. These are actions where the group need not be locally compact, or the action need not be strongly continuous. In particular, we consider the case where spectrum conditions are required for the implementing unitary group in covariant representations of such actions. The existence of a crossed product construction is guaranteed by the existence of "cross representations". For one-parameter automorphism groups, we prove that the existence of cross representations is stable with respect to a large set of perturbations of the action, and we fully analyze the structure of cross representations of inner actions on von Neumann algebras. For one-parameter automorphism groups we study the cross property for covariant representations, where the generator of the implementing unitary group is positive. In particular, we find that if the Borchers-Arveson minimal implementing group is cross, then so are all other implementing groups. For higher dimensional Lie group actions, we consider a class of spectral conditions which include the ones occurring in physics, and is sensible also for non-abelian or for infinite dimensional Lie groups. We prove that the cross property of a covariant representation is fully determined by the cross property of a certain one-parameter subsystem. This greatly simplifies the analysis of the existence of cross representations, and it allows us to prove the cross property for several examples of interest to physics. We also consider non-abelian extensions of the Borchers-Arveson theorem. There is a full extension in the presence of a cyclic invariant vector, but otherwise one needs to determine the vanishing of lifting obstructions.