Pontryagin Duality - Bohr Compactification and Almost-periodicity

Bohr Compactification and Almost-periodicity

One important application of Pontryagin duality is the following characterization of compact abelian topological groups:

Theorem. A locally compact abelian group G is compact if and only if the dual group G^ is discrete. Conversely, G is discrete if and only if G^ is compact.

That G being compact implies G^ is discrete or that G being discrete implies that G^ is compact is an elementary consequence of the definition of the compact-open topology on G^ and does not need Pontryagin duality. One uses Pontryagin duality to prove the converses.

The Bohr compactification is defined for any topological group G, regardless of whether G is locally compact or abelian. One use made of Pontryagin duality between compact abelian groups and discrete abelian groups is to characterize the Bohr compactification of an arbitrary abelian locally compact topological group. The Bohr compactification B(G) of G is H^, where H has the group structure G^, but given the discrete topology. Since the inclusion map

is continuous and a homomorphism, the dual morphism

is a morphism into a compact group which is easily shown to satisfy the requisite universal property.

See also almost periodic function.