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 $\alpha (\mathcal{A})$.

25 pages, to appear in the International Journal of Mathematics

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