# Pontryagin duality

A duality between topological groups and their character groups. The duality theorem states that if $G$ is a locally compact Abelian group and if $X ( G)$ is its character group, then the natural homomorphism $G \rightarrow X ( X ( G) )$ mapping an $a \in G$ to the character $\omega _ {a} : X ( G) \rightarrow T$, given by the formula

$$\omega _ {a} ( \alpha ) = \alpha ( a) ,\ \ \alpha \in X ( G) ,$$

is an isomorphism of topological groups. The following statements result from the above theorem.

I) If $H$ is a closed subgroup of $G$ and if

$$H ^ {*} = \{ {\alpha \in X ( G) } : {\alpha ( H) = 0 } \}$$

is its annihilator in $X ( G)$, then $H$ coincides with the annihilator

$$\{ {a \in G } : {\alpha ( a) = 0 \textrm{ for all } \ \alpha \in H ^ {*} } \}$$

of the subgroup $H ^ {*}$; moreover, the group $X ( H)$ is naturally isomorphic to $X ( G) / H ^ {*}$, and $X ( G/H)$ is isomorphic to the group $H ^ {*}$.

II) If $\phi : G \rightarrow H$ is a continuous homomorphism of locally compact Abelian groups, and $G$ is identified with $X ( X ( G) )$ and $H$ with $X ( X ( H) )$ by the natural isomorphisms, then the homomorphism $\phi$ can be identified with $( \phi ^ {*} ) ^ {*}$.

III) The weight of the group $X ( G)$( as a topological space, cf. Weight of a topological space) coincides with the weight of the group $G$.

Pontryagin duality establishes a correspondence between compact groups $G$ and discrete groups $X ( G)$, and vice versa. Moreover, a compact group $G$ is connected if and only if $X ( G)$ is torsion-free. A compact group $G$ is of dimension $n < \infty$ if and only if $X ( G)$ has finite rank $n$( see Rank of a group). A compact group $G$ is locally connected if and only if every finite-rank pure subgroup of $X ( G)$ is free. For finite groups $G$, Pontryagin duality coincides with duality between finite Abelian groups considered over the field $\mathbf C$ of complex numbers.

Topological groups for which the duality theorem is valid are called reflexive. Locally compact groups are not the only reflexive groups, since any reflexive Banach space, regarded as a topological group, is reflexive . On the characterization of reflexive groups, see .

There is an analogue of Pontryagin duality for non-commutative groups (the duality theorem of Tannaka–Krein) (see , , ). Let $G$ be a compact topological group, let $R$ be the algebra of complex-valued functions on $G$ whose translates span a finite-dimensional vector space and let $S ( R)$ be the set of all non-zero algebra homomorphisms $\omega : R \rightarrow \mathbf C$ satisfying the condition $\omega ( \overline{f}\; ) = \overline{ {\omega ( f ) }}\;$, $f \in R$. One can define a multiplication on $S ( R)$ which makes $S ( R)$ into a topological group with respect to the topology of pointwise convergence. To each $g \in G$ corresponds the homomorphism $\alpha _ {g} \in S ( R)$ given by the formula

$$\alpha _ {g} ( f ) = f ( g) ,\ \ f \in R .$$

Then the correspondence $g \rightarrow \alpha _ {g}$ is an isomorphism of the topological group $G$ onto $S ( R)$. There is also an algebraic description of the category of algebras $R$, which thus turns out to be dual to the category of compact topological groups. This theory admits a generalization to the case of homogeneous spaces of compact topological groups (see ).

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How to Cite This Entry:
Pontryagin duality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pontryagin_duality&oldid=49720
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article