On finitely summable Fredholm modules from Smale spaces

Abstract

We prove that all K K -homology classes of the stable (and unstable) Ruelle algebra of a Smale space have explicit Fredholm module representatives that are finitely summable on the same smooth subalgebra and with the same degree of summability. The smooth subalgebra is induced by a metric on the underlying Smale space groupoid and fine transversality relations between stable and unstable sets. The degree of summability is related to the fractal dimension of the Smale space. Further, the Fredholm modules are obtained by taking Kasparov products with a fundamental class of the Spanier-Whitehead K K -duality between the Ruelle algebras. Finally, we obtain general results on stability under holomorphic functional calculus and construct Lipschitz algebras on étale groupoids.

Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers

“…in the book [HiR00]) relying on the notion of «Fredholm module» and we need such a functional calculus to construct non-trivial K-homology classes from the Banach spectral triples obtained in this paper. We refer to [FGMR19], [Ger22], [GuS23] and [AGN24] for some recent papers on (classical) K-homology. Now, we introduce the following condition.…”