# Spectral triples on $O_N$

Research paper by Magnus Goffeng, Bram Mesland

Indexed on: 05 Oct '16Published on: 05 Oct '16Published in: arXiv - Mathematics - Operator Algebras

#### Abstract

We give a construction of an odd spectral triple on the Cuntz algebra $O_{N}$, whose class generates $K^1(O_{N})$. 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.