Non-Commutative Harmonic Analysis

For present purposes, we shall define non-commutative harmonic analysis to mean the decomposition of functions on a locally compact G-space X,1 where G is some (locally compact) group, into functions well-behaved with respect to the action of G. The classical cases are of course Fourier series, when G = X = T, the circle group, and the Fourier transform, when G = X = R, but we will mostly be concerned with the case when G is non-commutative. Since this subject is inextricably linked with the subject of representations of G (unitary representations, if we specialize to the case of L2-functions), we will also consider the general theory of representations of locally compact groups and of various related structures, such as Lie algebras and Jordan algebras.