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arXiv · 2102.05392

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

Valeriano Aiello·Daniele Guido·Tommaso Isola
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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 $\alpha(\mathcal A)$.

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Valeriano Aiello, Daniele Guido, Tommaso Isola. 2021-02-10. Spectral triples on irreversible $C^*$-dynamical systems. https://doi.org/10.1142/s0129167x22500057

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