Generalization of a Theorem on the Equivalence of the Coordinate and Algebraic Definitions of a Smooth Manifold

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...

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Published inJournal of mathematical sciences (New York, N.Y.) Vol. 263; no. 5; pp. 710 - 712
Main Author Krein, M. N.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.06.2022
Springer
Springer Nature B.V
Subjects
Algebra
Equivalence
Homomorphisms
Manifolds
Mathematical analysis
Mathematics
Mathematics and Statistics
Real numbers
Theorems
47A99
manifold
associative algebra
central algebra
homomorphism of algebras
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ISSN1072-3374
1573-8795
DOI10.1007/s10958-022-05961-2

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Summary: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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ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-022-05961-2