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Journal of Operator Theory

Volume 76, Issue 2, Fall 2016  pp. 307-335.

Ergodic actions and spectral triples

Authors:  Olivier Gabriel (1) and Martin Grensing (2)
Author institution:(1) Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, K-2100 Copenhagen O, Denmark
(2) MAPMO, Universite d'Orleans, B.P. 6759, 45 067 Orleans cedex 2, France

Summary: In this article, we give a general construction of spectral triples from certain Lie group actions on unital $C^*$-algebras. If the group $G$ is compact and the action is ergodic, we actually obtain a real and finitely summable spectral triple which satisfies the first order condition of Connes' axioms. This provides a link between the ``algebraic'' existence of ergodic action and the ``analytic'' finite summability property of the unbounded selfadjoint operator. More generally, for compact $G$ we carefully establish that our (symmetric) unbounded operator is essentially selfadjoint. Our results are illustrated by a host of examples --- including noncommutative tori and quantum Heisenberg manifolds.

Keywords: spectral triple, Lie group, ergodic action, Dirac operator, $K$-homology, unbounded Fredholm module

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