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Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory, 2017, Volume 129, Pages 3–133
(Mi into150)
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This article is cited in 3 scientific papers (total in 3 papers)
Continuous and smooth envelopes of topological algebras. Part 1
S. S. Akbarov All-Russian Institute for Scientific and Technical Information of Russian Academy of Sciences
Abstract:
Since the first optical instruments have been invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way of formalizing this idea in mathematics is the construction that assigns to an
arbitrary object $A$ in a category $K$ its envelope $\operatorname{Env}^{\Omega}_{\Phi}A$ in a given class of morphisms (a class
of representations) with respect to a given class of morphisms (a class of observation tools) $\Phi$. It turns out that if we take a sufficiently wide category of topological algebras as $K$, then each choice of the classes $\Omega$ and $\Phi$ defines a “projection of functional analysis into geometry”, and the standard “geometric disciplines”, like complex geometry, differential geometry, and topology, become special cases of this construction. This gives a formal scheme of “categorical construction of geometries” with many interesting applications, in particular, “geometric generalizations of the Pontryagin duality” (to the classes of noncommutative groups). In this paper, we describe this scheme in topology and in differential geometry.
Keywords:
stereotype space, stereotype algebra, envelope, Pontryagin duality.
Citation:
S. S. Akbarov, “Continuous and smooth envelopes of topological algebras. Part 1”, Functional analysis, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 129, VINITI, Moscow, 2017, 3–133; J. Math. Sci. (N. Y.), 227:5 (2017), 531–668
Linking options:
https://www.mathnet.ru/eng/into150 https://www.mathnet.ru/eng/into/v129/p3
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Abstract page: | 438 | Full-text PDF : | 174 | References: | 17 | First page: | 22 |
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