Metric and spectral triples for Dirichlet and resistance forms

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.

Densely defined non‐closable curl on carpet‐like metric measure spaces

“…As m is fixed, we write for the densities . Remark Such and m always exist: By folklore arguments we can find an energy dominant measure m and a countable family of functions that is dense in , separates the points of X and is such that , see for instance [, Lemma 2.1]. Let denote the algebra generated by and the constants.…”

“…We consider the Hilbert space of L 2 ‐ differential 1‐forms associated with and the corresponding first order derivation as introduced in and studied in . This is a generalized L 2 ‐theory of 1‐forms.…”

“…Our paper is a part of a broader program that aims to connect research on derivatives on fractals ( [8,9,[13][14][15][25][26][27][28]31,33,35,[37][38][39][40][41]46,51,54] and references therein) and on more general regular Dirichlet spaces [29,30,34] with classical and geometric analysis on metric measure spaces ( [6,10,12,[21][22][23][24][42][43][44][45]50] and references therein). In our previous article [36] we showed that on certain topologically one-dimensional spaces with a strongly local regular Dirichlet form one can prove a natural version of the Hodge theorem for 1-forms defined in 2 -sense: the set of harmonic 1-forms is dense in the orthogonal complement of the exact 1-forms.…”

Spectral triples for nested fractals

“…The idea of constructing a spectral triple on a fractal as a countable direct sum of finite dimensional spectral triples has been extended from fractals to general compact metric spaces in [33], where the Hausdorff dimension and measure, and the metric were recovered from their noncommutative analogues. More recently,in [20] a different approach to constructing spectral triples on metric spaces was taken, based on [10,11], so the starting point is a regular symmetric Dirichlet form on a locally compact separable metric space, endowed with a nonnegative Radon measure, and the intrinsic (or Carnot-Carathéodory) metric is recovered.…”

“…The idea of constructing a spectral triple on a fractal as a countable direct sum of finite dimensional spectral triples has been extended from fractals to general compact metric spaces in [33], where the Hausdorff dimension and measure, and the metric were recovered from their noncommutative analogues. More recently,in [20] a different approach to constructing spectral triples on metric spaces was taken, based on [10,11], so the starting point is a regular symmetric Dirichlet form on a locally compact separable metric space, endowed with a nonnegative Radon measure, and the intrinsic (or Carnot-Carathéodory) metric is recovered.The basic requirements for a spectral triple T = (A, H, D), where A is a self-adjoint algebra of operators and D is an unbounded self-adjoint operator, both acting on the Hilbert space H, are the boundedness of the commutators [D, a], a ∈ A, and the compactness of the resolvents of D [13]. Based on these hypotheses, one may associate with…”