Noncommutative harmonic analysis

Summary

Noncommutative harmonic analysis is harmonic analysis on groups for locally compact groups that need not be abelian. The characters of the abelian theory are replaced by the irreducible unitary representations of the group. Its central questions are how the left regular representation on $L^2(G)$ splits into irreducible pieces and how a function is recovered from its Fourier transform. It covers the Peter–Weyl theorem for compact groups and the Plancherel theory of Lie groups such as $SL_2(\R)$.