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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.

DOI: http://dx.doi.org/10.7900/jot.2015sep25.2101
Keywords: spectral triple, Lie group, ergodic action, Dirac operator, K-homology, unbounded Fredholm module


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