Pontryagin duality

In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. It places in a unified context a number of observations about functions on the real line or on finite abelian groups:

Non-commutative theory

Such a theory cannot exist in the same form for non-commutative groups G, since in that case the appropriate dual object G^ of isomorphism classes of representations cannot only contain one-dimensional representations, and will fail to be a group. The generalisation that has been found useful in category theory is called Tannaka-Krein duality; but this diverges from the connection with harmonic analysis, which needs to tackle the question of the Plancherel measure on G^.

Related Topics:
Category theory - Tannaka-Krein duality - Harmonic analysis - Plancherel

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There are analogues of duality theory for noncommutative groups, some of which are formulated in the language of C*- algebras.

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