Spectral triples on $O_N$

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.

15 pages. To appear in the conference proceedings from the MATRIX-program C*-algebraic invariants for dynamics using KK-theory in Creswick, Australia, 2016