Spectral triples and finite summability on Cuntz–Krieger algebras

Abstract

We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz–Krieger algebras by means of realizing these algebras as “the algebra of functions on a non-commutative space” coming from a sub shift of finite type. We show that any odd $K$-homology class can be represented by such an odd bounded Fredholm module or odd spectral triple. The odd bounded Fredholm modules that are constructed are finitely summable. The spectral triples are $\theta$-summable, although their phases will already on the level of analytic $K$-cycles be finitely summable bounded Fredholm modules. Using the unbounded Kasparov product, we exhibit a family of generalized spectral triples, related to work of Bellissard–Pearson, possessing mildly unbounded commutators, whilst still giving well defined $K$-homology classes.