Spectral triples on irreversible $C^*$-dynamical systems
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 $\alpha(\mathcal A)$.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2021
- DOI:
- 10.48550/arXiv.2102.05392
- arXiv:
- arXiv:2102.05392
- Bibcode:
- 2021arXiv210205392A
- Keywords:
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- Mathematics - Operator Algebras;
- 58B34;
- 46LXX;
- 47L65
- E-Print:
- 25 pages, to appear in the International Journal of Mathematics