Metrics and spectral triples for Dirichlet and resistance forms
arXiv:1309.5937 · doi:10.4171/JNCG/195
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.
References in corpus (5)
Cited by in corpus (14)
- Dirac and magnetic Schrödinger operators on fractals
- Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals
- Energy and Laplacian on Hanoi-type fractal quantum graphs
- Singularly continuous spectrum of a self-similar Laplacian on the half-line
- Fractal snowflake domain diffusion with boundary and interior drifts
- Fractal AC circuits and propagating waves on fractals
- Finite energy coordinates and vector analysis on fractals
- Differential forms on Dirichlet spaces and Bakry-Émery estimates on metric graphs
- Canonical diffusions on the pattern spaces of aperiodic Delone sets
- Examples of infinite direct sums of spectral triples
- From non-symmetric particle systems to non-linear PDEs on fractals
- Gaps in the spectrum of the Laplacian on -Gaskets
- Differential forms for fractal subspaces and finite energy coordinates
- Dual graphs and modified Barlow--Bass resistance estimates for repeated barycentric subdivisions