[1] Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier transforms, the major business of non-commutative harmonic analysis is usually taken to be the extension of the theory to all groups G that are locally compact.
These examples are of interest and frequently applied in mathematical physics, and contemporary number theory, particularly automorphic representations.
What to expect is known as the result of basic work of John von Neumann.
It is parametrized therefore by the unitary dual, the set of isomorphism classes of such representations, which is given the hull-kernel topology.
The further theory divides up the Plancherel measure into a discrete and a continuous part.