Fourier transform on a nonabelian group

Summary

The Fourier transform on a nonabelian locally compact group sends a function on the group to a family of operators. There is one operator for each irreducible unitary representation. For an abelian group the irreducible unitary representations are characters and the operators are numbers. One then gets back the Fourier transform on locally compact abelian groups. It turns convolution into composition of operators. The Plancherel theorem for groups and the Plancherel measure describe how it preserves the $L^2$ norm.