Spectral triples on $O_N$

TL;DR

This paper constructs a new spectral triple on the Cuntz algebra $O_N$, linking $K$-homology with a metric measure space structure and introducing a singular integral operator analogous to the Laplacian logarithm.

Contribution

It provides the first odd spectral triple on $O_N$ whose $K$-homology class generates $K^1(O_N)$, using a novel metric measure space approach and a singular integral operator.

Findings

Constructed an odd spectral triple on $O_N$

Introduced a singular integral operator analogous to the Laplacian logarithm

Achieved a $ heta$-summable spectral triple with finitely summable phase

Abstract

We give a construction of an odd spectral triple on the Cuntz algebra $O_{N}$, whose $K$-homology class generates the odd $K$-homology group $K^1(O_{N})$. Using a metric measure space structure on the Cuntz-Renault groupoid, we introduce a singular integral operator which is the formal analogue of the logarithm of the Laplacian on a Riemannian manifold. Assembling this operator with the infinitesimal generator of the gauge action on $O_{N}$ yields a $\theta$-summable spectral triple whose phase is finitely summable. The relation to previous constructions of Fredholm modules and spectral triples on $O_{N}$ is discussed.