Spectral triples on irreversible $C^{\ast }$-dynamical systems

Abstract

Given a spectral triple on a $C^{\ast }$-algebra $\mathcal{A}$ together with a unital injective endomorphism $\alpha$, the problem of defining a suitable crossed product $C^{\ast }$-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 $\alpha (\mathcal{A})$.

arXiv comment25 pages, to appear in the International Journal of Mathematics

References in corpus (13)

• Metrics on State Spaces
• Spectral triples for AF C*-algebras and metrics on the Cantor set
• Dimensions and singular traces for spectral triples, with applications to fractals
• Dynamical Systems on Spectral Metric Spaces
• Crossed products by endomorphisms and reduction of relations in relative Cuntz-Pimsner algebras
• Dynamics of compact quantum metric spaces
• Wieler solenoids, Cuntz-Pimsner algebras and K-theory
• Spectral Triples and Generalized Crossed Products
• Spectral triples for noncommutative solenoidal spaces from self-coverings
• Spectral triples for nested fractals
• Real dimensional spaces in noncommutative geometry
• A Spectral Triple for a Solenoid Based on the Sierpinski Gasket
• Every Continuum has a Compact Universal Cover