Editing Unitary representation (section)

==Context in harmonic analysis==

The theory of unitary representations of topological groups is closely connected with [[harmonic analysis]]. In the case of an [[abelian group]] ''G'', a fairly complete picture of the representation theory of ''G'' is given by [[Pontryagin duality]]. In general, the unitary equivalence classes (see [[#Formal definitions|below]]) of [[irreducible representation|irreducible]] unitary representations of ''G'' make up its '''unitary dual'''.  This set can be identified with the [[spectrum of a C*-algebra|spectrum of the C*-algebra]] associated with ''G'' by the [[group ring|group C*-algebra]] construction.  This is a [[topological space]].

The general form of the [[Plancherel theorem]] tries to describe the [[regular representation]] of ''G'' on ''L''<sup>2</sup>(''G'') using a [[measure (mathematics)|measure]] on the unitary dual. For ''G'' abelian this is given by the Pontryagin duality theory.  For ''G'' [[Compact group|compact]], this is done by the [[Peter–Weyl theorem]]; in that case, the unitary dual is a [[discrete space]], and the measure attaches an atom to each point of mass equal to its degree.