Journal of Mathematical Sciences

, Volume 227, Issue 6, pp 669–789 | Cite as

Continuous and Smooth Envelopes of Topological Algebras. Part 2

  • S. S. Akbarov


Since the first optical instruments were invented, the 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 \( {\mathrm{Env}}_{\varPhi}^{\varOmega}\kern0.5em A \) in a given class of morphisms (a class of representations) Ω with respect to a given class of morphisms (a class of observation tools) Φ. It turns out that if we take a sufficiently wide category of topological algebras as K, then each choice of the classes Ω and Φ 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.

Key words and phrases

stereotype space  stereotype algebra  envelope  Pontryagin duality 

AMS Subject Classification

46Hxx 54-xx  53-xx 


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Russian Institute for Scientific and Technical Information of the Russian Academy of SciencesMoscowRussia

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