  We develop a semigroup approach to representation theory for pro-Lie groups
satisfying suitable amenability conditions. As an application of our approach,
we establish a one-to-one correspondence between equivalence classes of unitary
irreducible representations and coadjoint orbits for a class of pro-Lie groups
including all connected locally compact nilpotent groups and arbitrary infinite
direct products of nilpotent Lie groups. The usual $C^*$-algebraic approach to
group representation theory positivey breaks down for infinite direct products
of non-compact locally compact groups, hence the description of their unitary
duals in terms of coadjoint orbits is particularly important whenever it is
available, being the only description known so far.
