Journal of Operator Theory

Volume 93, Issue 1, Winter 2025 pp. 147-171.

Quantum differentials of spectral triples, Dirichlet spaces and discrete group

Authors: Fabio E.G. Cipriani (1), Jean-Luc Sauvageot (2)
Author institution: (1) Politecnico di Milano, Dipartimento di Matematica, piazza Leonardo da Vinci 32, 20133 Milano, Italy
(2) Institut de Mathematiques de Jussieu -- Paris Rive Gauche, CNRS -- Universite Paris Cite, F-75205 Paris Cedex 13, France

Summary: We study natural conditions on essentially discrete spectral triples $(\mathcal{A},h,D)$ by which the quantum differential ${\bf da}$ of $a\in\mathcal{A}$ belongs to the ideal generated by the unit length ${\bf ds}=D^{-1}$. We also study upper and lower bounds on the singular values of the ${\bf da}$'s and apply the general framework to natural spectral triples of Dirichlet spaces and, in particular, to those on dual of discrete groups arising from negative definite functions.

DOI: http://dx.doi.org/10.7900/jot.2023jan04.2455
Keywords: spectral triple, quantum differential, singular value, Dirichlet form, discrete groups

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