Spectral triples for subshifts

TL;DR

This paper constructs spectral triples for algebras associated with subshifts, analyzing their properties and Connes' distance, providing new insights into the noncommutative geometry of dynamical systems.

Contribution

It introduces a novel construction of spectral triples on subshift algebras, extending previous ideas and analyzing their geometric and spectral properties.

Findings

Spectral triples can be constructed with bounded commutators for subshift algebras.

The Connes' distance can be either bounded or unbounded depending on the example.

The results suggest potential extensions to noncommutative algebras.

Abstract

We propose a construction for spectral triple on algebras associated with subshifts. One-dimensional subshifts provide concrete examples Z-actions on Cantor sets. The C*-algebra of this dynamical system is generated by functions in C(X) and a unitary element u implementing the action. Building on ideas of Christensen and Ivan, we give a construction of a family of spectral triples on the commutative algebra C(X). There is a canonical choice of eigenvalues for the Dirac operator D which ensures that [D,u] is bounded, so that it extends to a spectral triple on the crossed product. We study the summability of this spectral triple, and provide examples for which the Connes' distance associated with it on the commutative algebra is unbounded, and some for which it is bounded. We conjecture that our results on the Connes distance extend to the spectral triple defined on the noncommutative algebra.