  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.
