2018/09/25 by Hendrik Grundling, Grundling, Hendrik, Karl‐Hermann Neeb +1
Materials Science · Mathematics · #22F50 #46L40 #46L55 #46L60 (primary) #81R15 (secondary) #81T05 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Lanthanide and Transition Metal Complexes #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1809.09781
openalex publication_date 2018/09/25 · openalex created_date 2022/08/03 · openalex updated_date 2026/07/28
We analyze existence of crossed product constructions of Lie group actions on\nC^*-algebras which are singular. These are actions where the group need not be\nlocally compact, or the action need not be strongly continuous. In particular,\nwe consider the case where spectrum conditions are required for the\nimplementing unitary group in covariant representations of such actions. The\nexistence of a crossed product construction is guaranteed by the existence of\n"cross representations". For one-parameter automorphism groups, we prove that\nthe existence of cross representations is stable with respect to a large set of\nperturbations of the action, and we fully analyze the structure of cross\nrepresentations of inner actions on von Neumann algebras. For one-parameter\nautomorphism groups we study the cross property for covariant representations,\nwhere the generator of the implementing unitary group is positive. In\nparticular, we find that if the Borchers-Arveson minimal implementing group is\ncross, then so are all other implementing groups. For higher dimensional Lie\ngroup actions, we consider a class of spectral conditions which include the\nones occurring in physics, and is sensible also for non-abelian or for infinite\ndimensional Lie groups. We prove that the cross property of a covariant\nrepresentation is fully determined by the cross property of a certain\none-parameter subsystem. This greatly simplifies the analysis of the existence\nof cross representations, and it allows us to prove the cross property for\nseveral examples of interest to physics. We also consider non-abelian\nextensions of the Borchers-Arveson theorem. There is a full extension in the\npresence of a cyclic invariant vector, but otherwise one needs to determine the\nvanishing of lifting obstructions.\n