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Spectral triples on irreversible $C^*$-dynamical systems (2102.05392v2)
Published 10 Feb 2021 in math.OA
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 $\alpha(\mathcal A)$.