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arXiv · 2303.17903

Crossed products as compact quantum metric spaces

Mario Klisse
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Abstract

By employing the external Kasparov product, Hawkins, Skalski, White and Zacharias constructed spectral triples on crossed product C$^\ast$-algebras by equicontinuous actions of discrete groups. They further raised the question for whether their construction turns the respective crossed product into a compact quantum metric space in the sense of Rieffel. By introducing the concept of groups separated with respect to a given length function, we give an affirmative answer in the case of virtually Abelian groups equipped with certain orbit metric length functions. We further complement our results with a discussion of natural examples such as generalized Bunce-Deddens algebras and higher-dimensional non-commutative tori.

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Mario Klisse. 2023-03-31. Crossed products as compact quantum metric spaces. https://doi.org/10.4153/s0008414x24000877

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