Pontryagin duality
The subject is named after Lev Pontryagin who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934.
This was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940.
Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups: The theory, introduced by Lev Pontryagin and combined with the Haar measure introduced by John von Neumann, André Weil and others depends on the theory of the dual group of a locally compact abelian group.
Examples of locally compact abelian groups include finite abelian groups, the integers (both for the discrete topology, which is also induced by the usual metric), the real numbers, the circle group T (both with their usual metric topology), and also the p-adic numbers (with their usual p-adic topology).
This is strongly analogous to the canonical isomorphism between a finite-dimensional vector space and its double dual,
is a countably additive measure μ defined on the Borel sets of
and also satisfies some regularity conditions (spelled out in detail in the article on Haar measure).
that appears in the Fourier inversion formula is called the dual measure to
is the Lebesgue measure on Euclidean space, we obtain the ordinary Fourier transform on
The space of integrable functions on a locally compact abelian group
The Fourier transform takes convolution to multiplication, i.e. it is a homomorphism of abelian Banach algebras
corresponds a unique multiplicative linear functional on the group algebra defined by
is a morphism into a compact group which is easily shown to satisfy the requisite universal property.
[6] In particular, Samuel Kaplan[7][8] showed in 1948 and 1950 that arbitrary products and countable inverse limits of locally compact (Hausdorff) abelian groups satisfy Pontryagin duality.
More recently, Sergio Ardanza-Trevijano and María Jesús Chasco[10] have extended the results of Kaplan mentioned above.
-spaces but not necessarily locally compact, provided some extra conditions are satisfied by the sequences.
However, there is a fundamental aspect that changes if we want to consider Pontryagin duality beyond the locally compact case.
is a Hausdorff abelian topological group that satisfies Pontryagin duality, and the natural evaluation pairing
As a corollary, all non-locally compact examples of Pontryagin duality are groups where the pairing
In the 1990s Sergei Akbarov[16] gave a description of the class of the topological vector spaces that satisfy a stronger property than the classical Pontryagin reflexivity, namely, the identity
endowed with the topology of uniform convergence on totally bounded sets in
The spaces of this class are called stereotype spaces, and the corresponding theory found a series of applications in Functional analysis and Geometry, including the generalization of Pontryagin duality for non-commutative topological groups.
the classical Pontryagin construction stops working for various reasons, in particular, because the characters don't always separate the points of
At the same time it is not clear how to introduce multiplication on the set of irreducible unitary representations of
, and it is even not clear whether this set is a good choice for the role of the dual object for
) into the category of finite dimensional Hopf algebras, so that the Pontryagin duality functor
of taking the dual vector space (which is a duality functor in the category of finite dimensional Hopf algebras).
[20] In 1973 Leonid I. Vainerman, George I. Kac, Michel Enock, and Jean-Marie Schwartz built a general theory of this type for all locally compact groups.
[21] From the 1980s the research in this area was resumed after the discovery of quantum groups, to which the constructed theories began to be actively transferred.
[20] This deficiency can be corrected (for some classes of groups) within the framework of duality theories constructed on the basis of the notion of envelope of topological algebra.
