Abstract
In this paper, we generalize the theorem on the equivalence of the coordinate and algebraic definitions of a smooth manifold. Within the framework of the algebraic approach, a point is considered as a homomorphism from the algebra of smooth real functions defined on a manifold into the field of real numbers. We consider a generalization for the case where the field of real numbers is replaced by an arbitrary associative normalized algebra, generally speaking, noncommutative.
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Reference
J. Nestruev, Smooth Manifolds and Observables, Springer-Verlag, New York–Berlin–Heidelberg (2003).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 171, Proceedings of the Voronezh Winter Mathematical School “Modern Methods of Function Theory and Related Problems,” Voronezh, January 28 – February 2, 2019. Part 2, 2019.
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Krein, M.N. Generalization of a Theorem on the Equivalence of the Coordinate and Algebraic Definitions of a Smooth Manifold. J Math Sci 263, 710–712 (2022). https://doi.org/10.1007/s10958-022-05961-2
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DOI: https://doi.org/10.1007/s10958-022-05961-2