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Journal of Operator Theory

Volume 84, Issue 2, Fall 2020  pp. 369-451.

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

Authors:  Hendrik Grundling (1), Karl-Hermann Neeb (2)
Author institution: (1) Department of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
(2) Department of Mathematics, FAU Erlangen- Nurnberg, Cauerstrasse 11, 91058 Erlangen, Germany

Summary: We analyze existence of crossed product constructions for singular group actions on $C^*$-algebras, i.e. where the group need not be locally compact, or the action need not be strongly continuous. This is specialized to the case where spectrum conditions are required for the implementing unitary groups in covariant representations. The existence of a crossed product construction is guaranteed by the existence of "cross representations". For one-parameter automorphism groups, this existence property is stable with respect to many perturbations of the action. The structure of cross representations of inner actions on von Neumann algebras is obtained. We analyze the cross property for covariant representations of one-parameter automorphism groups, where the generator of the implementing unitary group is positive. If the Borchers-Arveson minimal implementing group is cross, then so are all other implementing groups.s This analysis is extended here to higher dimensional Lie group actions, including several examples of interest to physics.

Keywords: $C^*$-algebra, group algebra, crossed product, topological group, singular action, spectral condition, cross representation

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