  This is the third paper of a series relating the equivariant twisted
$K$-theory of a compact Lie group $G$ to the ``Verlinde space'' of isomorphism
classes of projective lowest-weight representations of the loop groups. Here,
we treat arbitrary compact Lie groups. In addition, we discuss the relation to
semi-infinite cohomology, the fusion product of Conformal Field theory, the
r\^ole of energy and the topological Peter-Weyl theorem.
