Pontryagin Duality in the Theory of Topological Vector Spaces and in Topological Algebra

  title={Pontryagin Duality in the Theory of Topological Vector Spaces and in Topological Algebra},
  author={S. S. Akbarov},
  journal={Journal of Mathematical Sciences},
  • S. Akbarov
  • Published 2003
  • Mathematics
  • Journal of Mathematical Sciences
The theory of topological vector spaces (TVS), being a foundation of modern functional analysis, is now considered as a completely mature, or, to be more specific, dead mathematical discipline. This pessimistic view is based on a picture, where, on the one hand, a well-known system of facts is stated, facts that have been considered classical since the times of Mackey and Grothendieck, and, on the other hand, an opinion exists explicitly or implicitly, that any deviation from this system… 

Pontryagin duality and topological algebras

As is known, the Pontryagin duality is a traditional object of investigations in Group Theory. Another application of this idea was found quite recently in the theory of topological algebras and is

Continuous and Smooth Envelopes of Topological Algebras. Part 2

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

Envelopes and refinements in categories, with applications to functional analysis

An envelope in a category is a construction that generalizes the operations of "exterior completion", like completion of a locally convex space, or Stone-\v{C}ech compactification of a topological

Two classes of spaces reflexive in the sense of Pontryagin

The Pontryagin-van Kampen duality for locally compact Abelian groups can be generalized in two ways to wider classes of topological Abelian groups: in the first approach the dual group is endowed

Continuous and smooth envelopes of topological algebras

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.

An algebra of continuous functions as a continuous envelope of its subalgebras

To an arbitrary involutive stereotype algebra A the continuous envelope operation assigns its nearest, in some sense, involutive stereotype algebra EnvCA so that homomorphisms to various C*-algebras

On continuous duality for Moore groups

  • S. Akbarov
  • Mathematics
    Journal of Operator Theory
  • 2022
n 2013, Yu.N.~Kuznetsova constructed a duality theory for Moore groups, based on the idea of a continuous envelope of topological algebra and having the advantage over the existing theories that its

Stereotype approximation property for the group algebra ${\mathcal C}^\star(G)$ of measures

In [1, 3] the author described the stereotype approximation property, an analog of the classical approximation property transferred into the category Ste of stereotype spaces. It was noticed in [3]

Duality for SIN-groups

We suggest a generalization of Pontryagin duality from the category of Abelian locally compact groups to a category which includes all SIN-groups, i.e. groups with a base of invariant neighborhoods

Holomorphic duality for countable discrete groups

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



Convenient categories of topological algebras

Introduction. Concrete associative algebras with a topology have long arisen in mathematical practice; thus, a notion of topological space with algebraic operations making it an associative algebra

The Pontrjagin Duality Theorem in Linear Spaces

The Pontrjagin Duality Theorem, known to be true for locally compact groups, asserts that the given group and the character group of its character group are isomorphic under a "natural" mapping.

Topological Vector Spaces

The general theory of topological vector spaces was founded during the period which goes from 1920 to 1930 approximately. But it had been prepared for a long time before by the study of numerous


  • G. M. Kelly
  • Mathematics
    Elements of ∞-Category Theory
  • 2005
Although numerous contributions from divers authors, over the past fifteen years or so, have brought enriched category theory to a developed state, there is still no connected account of the theory,

A guide to quantum groups

Introduction 1. Poisson-Lie groups and Lie bialgebras 2. Coboundary Poisson-Lie groups and the classical Yang-Baxter equation 3. Solutions of the classical Yang-Baxter equation 4. Quasitriangular

Stereotype locally convex spaces

We give complete proofs of some previously announced results in the theory of stereotype (that is, reflexive in the sense of Pontryagin duality) locally convex spaces. These spaces have important

Representation theory of finite groups and associated algebras

A loom sley capable of use in high speed looms comprises a substantially tubular member, said tubular member having attached thereto a race board plate and a reed support device, said race board

Foundations of Differentiable Manifolds and Lie Groups

1 Manifolds.- 2 Tensors and Differential Forms.- 3 Lie Groups.- 4 Integration on Manifolds.- 5 Sheaves, Cohomology, and the de Rham Theorem.- 6 The Hodge Theorem.- Supplement to the Bibliography.-

Abstract Harmonic Analysis

The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria.