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arXiv · 1309.5937

Metrics and spectral triples for Dirichlet and resistance forms

Michael Hinz·Daniel J. Kelleher·Alexander Teplyaev
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Abstract

The article deals with intrinsic metrics, Dirac operators and spectral triples induced by regular Dirichlet and resistance forms. We show, in particular, that if a local resistance form is given and the space is compact in resistance metric, then the intrinsic metric yields a geodesic space. Given a regular Dirichlet form, we consider Dirac operators within the framework of differential 1-forms proposed by Cipriani and Sauvageot, and comment on its spectral properties. If the Dirichlet form admits a carré operator and the generator has discrete spectrum, then we can construct a related spectral triple, and in the compact and strongly local case the associated Connes distance coincides with the intrinsic metric. We finally give a description of the intrinsic metric in terms of vector fields.

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Michael Hinz, Daniel J. Kelleher, Alexander Teplyaev. 2013-10-07. Metrics and spectral triples for Dirichlet and resistance forms. https://doi.org/10.4171/jncg%2F195

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