Pontryagin duality theorem - Imported Knowlpedia Content

Statement

Let $G$ be a locally compact Hausdorff group that is abelian, and let $\widehat G$ be its Pontryagin dual, both duals carrying the compact-open topology. The Pontryagin duality theorem states that the evaluation homomorphism

\eta_G:G\longrightarrow\widehat{\widehat G}, \qquad \eta_G(x)(\gamma)=\gamma(x),

is an isomorphism of groups and a homeomorphism. Hence the characters of $G$ separate its points, every continuous character of $\widehat G$ is evaluation at a unique point of $G$ , and the topology of $G$ is recovered by the bidual.

Hypotheses and naturality

Local compactness and the compact-open topology are essential parts of the theorem. The maps $\eta_G$ are natural: for a continuous homomorphism $u:G\to H$ , dualization gives $\widehat u:\widehat H\to\widehat G$ by precomposition, and the resulting bidual map satisfies

\widehat{\widehat u}\circ\eta_G=\eta_H\circ u.

Thus Pontryagin duality is a contravariant equivalence on the category of locally compact abelian groups Rudin, §1.7.

Compact-discrete correspondence

Duality exchanges compact and discrete groups. If $G$ is compact, $\widehat G$ is discrete; if $G$ is discrete, $\widehat G$ is compact. It also exchanges closed subgroups with annihilator quotients, turning exact sequences into reversed exact sequences under the standard closedness hypotheses.

Examples and limitations

The canonical evaluation maps identify

\widehat{\widehat{\mathbb R}}\cong\mathbb R,\qquad \widehat{\widehat{\mathbb Z}}\cong\mathbb Z,\qquad \widehat{\widehat{\mathbb T}}\cong\mathbb T.

Finite abelian groups are abstractly isomorphic to their duals, but such a self-duality need not be canonical; the canonical statement is the bidual isomorphism.

References

W. Rudin, Fourier Analysis on Groups, Wiley-Interscience, 1962. DOI record. Relevant: Chapter 1, the Pontryagin duality theorem.
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Volume I, Springer, 1963. DOI record. Relevant: character groups and duality for locally compact abelian groups.