On the Unitary Representation Theory of Locally Compact Contraction Groups
Copyright policy )On the Unitary Representation Theory of Locally Compact Contraction Groups
The unitary representation theory of locally compact contraction groups and their semi-direct products with $\mathbb{Z}$ is studied. We put forward the problem of completely characterising such groups which are type I or CCR and this article provides a stepping stone towards a solution to this problem. In particular, we determine new examples of type I and non-type-I groups in this class, and we completely classify the irreducible unitary representations of the torsion-free groups, which are shown to be type I. When these groups are totally disconnected, they admit a faithful action by automorphisms on an infinite locally-finite regular tree; this work thus provides new examples of automorphism groups of regular trees with interesting representation theory, adding to recent work on this topic.
48 pages. To appear in the Journal of Lie Theory
Representation theory for linear algebraic groups, amenable group, \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations, Mathematics - Operator Algebras, contraction group, Unitary representations of locally compact groups, Group Theory (math.GR), Other representations of locally compact groups, type \(I\) group, 20C25, 20G05, 22D10, 22D12, 22D25, 43A65, CCR group, groups acting on trees, unipotent linear algebraic group, scale group, FOS: Mathematics, Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis), Representation Theory (math.RT), Projective representations and multipliers, Operator Algebras (math.OA), Mathematics - Group Theory, Mathematics - Representation Theory, unitary representation
Representation theory for linear algebraic groups, amenable group, \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations, Mathematics - Operator Algebras, contraction group, Unitary representations of locally compact groups, Group Theory (math.GR), Other representations of locally compact groups, type \(I\) group, 20C25, 20G05, 22D10, 22D12, 22D25, 43A65, CCR group, groups acting on trees, unipotent linear algebraic group, scale group, FOS: Mathematics, Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis), Representation Theory (math.RT), Projective representations and multipliers, Operator Algebras (math.OA), Mathematics - Group Theory, Mathematics - Representation Theory, unitary representation
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