arXiv subjects
Publications and source records attributed to S. S. Akbarov.
This book contains the material of my research on stereotype duality theories in geometry. It was intended as a continuation of my recently published monograph in De Gruyter on stereotype spaces and algebras.
In 2008, the author proposed a version of duality theory for (not necessarily, Abelian) complex Lie groups, based on the idea of using the Arens-Michael envelope of topological algebra and having an advantage over existing theories in that the enclosing category in it consists of Hopf algebras in the classical sense. Recently these results were refined and corrected by O.Yu.Aristov. In this paper, we propose a generalization of this theory to the class of (not necessarily Abelian) countable discrete groups.
We prove two identities that connect some natural tensor products in the category $\sf{LCS}$ of locally convex spaces with the tensor products in the category $\sf{Ste}$ of stereotype spaces. In particular, we give sufficient conditions under which the identity $$ X^\vartriangle\odot Y^\vartriangle\cong (X^\vartriangle\cdot Y^\vartriangle)^\vartriangle\cong (X\cdot Y)^\vartriangle $$ holds, where $\odot$ is the injective tensor product in the category $\sf{Ste}$, $\cdot$, the primary tensor product in the category $\sf{LCS}$, and $\vartriangle$, the pseudosaturation operation in the category $\sf{LCS}$. Studying the relations of this type is justified by the fact that they turn out to be important instruments for constructing duality theory based on the notion of envelope. In particular, they are used in the construction of the duality theory for the class of (not necessarily, Abelian) countable discrete groups.
We prove several properties of kernels and cokernels in the category of augmented involutive stereotype algebras: 1) the morphisms of the augmented involutive stereotype algebras have kernels and cokernels, 2) the cokernel is preserved under the passage to the group stereotype algebras, and 3) the notion of cokernel allows to prove that the continuous envelope $\operatorname{Env} {\mathcal C}^\star(G)$ of the group algebra ${\mathcal C}^\star(G)$ is an involutive Hopf algebra in the category of stereotype spaces $({\tt Ste},\odot)$, if $G$ has the form $Z\cdot K$, where $Z$ is a commutative locally compact group, and $K$ a compact group. The last result plays an important role in the generalization of the Pontryagin duality for arbitrary Moore groups.
In this paper we correct the errors of Yu.~N.~Kuznetsova's paper on the continuous duality for Moore groups.
The stereotype approximation property is formally a stronger condition than the classical approximation property, and because of that the question which spaces possess the stereotype approximation property is quite difficult. In this paper we show that the group algebra ${\mathcal C}^\star(G)$ of measures on a locally compact grop $G$ always has this property.
Since the time when 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 to formalize it in mathematics is the construction that assigns to an arbitrary object $A$ in a category $K$ its envelope $\text{Env}_\varPhi^\varOmega A$ in a given class of morphisms (a class of representations) $\varOmega$ with respect to a given class of morphisms (a class of observation tools) $\varPhi$. It turns out that if we take a sufficiently wide category of topological algebras as $K$, then each choice of the classes $\varOmega$ and $\varPhi$ 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 non-commutative groups). In this paper we describe this scheme in topology and in differential geometry.
We suggest a generalization of Pontryagin duality from the category of commutative Stein groups to the category of (not necessarily commutative) Stein groups with algebraic connected component of identity. In contrast to the other similar generalizations, in our approach the enveloping category consists of Hopf algebras (in a proper symmetrical monoidal category).