Khalkhali, M., & Tageddine, D. (2024). Noncommutative geometry on the Berkovich projective line. arXiv preprint arXiv:2411.02593.
This paper aims to demonstrate that non-Archimedean geometries, specifically the Berkovich projective line (P1Berk(Cp)), offer natural examples of noncommutative geometries. The authors achieve this by constructing and analyzing several C*-algebras and spectral triples associated with P1Berk(Cp).
The authors utilize various mathematical tools and concepts, including:
The study provides compelling evidence that the Berkovich projective line exhibits characteristics of noncommutative geometry. The constructed C*-algebras and spectral triples offer valuable tools for investigating the geometric and arithmetic aspects of P1Berk(Cp).
This research significantly contributes to the understanding of noncommutative geometry in the context of non-Archimedean spaces. It opens up new avenues for exploring the interplay between these areas and their applications in other mathematical fields.
The paper primarily focuses on the Berkovich projective line as a foundational example. Further research could explore the extension of these constructions and analyses to higher-dimensional Berkovich spaces or other non-Archimedean settings. Additionally, investigating the potential connections between these noncommutative geometric structures and other arithmetic or geometric invariants associated with P1Berk(Cp) would be of interest.
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by Masoud Khalk... at arxiv.org 11-06-2024
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