Spectral triples on irreversible $C^*$-dynamical systems

TL;DR

This paper develops a method to construct spectral triples on crossed product $C^*$-algebras arising from irreversible dynamical systems, extending noncommutative geometry tools to new classes of non-invertible actions.

Contribution

It introduces a novel construction of spectral triples on crossed products of $C^*$-algebras with endomorphisms, based on expansive conditions and dual covering projections.

Findings

Provides a framework for spectral triples on non-invertible dynamical systems.

Connects the expansiveness of endomorphisms with Lip-norm compatibility.

Extends noncommutative geometric analysis to irreversible $C^*$-dynamics.

Abstract

Given a spectral triple on a $C^*$-algebra $\mathcal A$ together with a unital injective endomorphism $\alpha$, the problem of defining a suitable crossed product $C^*$-algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and of Hawkins, Skalski, White and Zacharias, and on our previous papers. The embedding of $\alpha(\mathcal A)$ in $\mathcal A$ can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricity of the covering projection and is expressed via the compatibility of the Lip-norms on $\mathcal A$ and $\alpha(\mathcal A)$.