Quantum differentials of Spectral triples, Dirichlet spaces and discrete groups

TL;DR

This paper investigates conditions under which quantum differentials in spectral triples relate to the Dirac operator, providing bounds and applying the framework to Dirichlet spaces and discrete groups.

Contribution

It introduces conditions linking quantum differentials to the Dirac operator and applies these to spectral triples on Dirichlet spaces and discrete groups.

Findings

Quantum differentials belong to specific ideals under certain conditions.

Bounds on singular values of quantum differentials are established.

Applications to spectral triples on Dirichlet spaces and duals of discrete groups.

Abstract

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