Harmonic analysis on unitary groups

A theory of harmonic analysis on a metric group (G, d) is developed with the model of U$\text{U}$, the unitary group of a C¹-algebra $\text{U}$, in mind. Essential in this development is the set $\text{G}\text{?}{}_{\mathrm{d}}$ of contractive, irreducible representations of G, and its concomitant set P_d(G) of positive-definite functions. It is shown that $\text{G}\text{?}{}_{\mathrm{d}}$ is compact and closed in $\text{G}\text{?}$. The set $\text{G}\text{?}{}_{\mathrm{d}}$ is determined in a number of cases, in particular when G = U($\text{U}$) with $\text{U}$ abelian. If $\text{U}$ is an AW¹-algebra, it is shown that $\text{G}\text{?}$_d is essentially the same as $\text{U}\text{?}$. Unitary groups are characterised in terms of a certain Lie algebra $\text{g}$_u and several characterisations of G = U($\text{U}$) when $\text{U}$ is abelian are given.