2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99917This 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 rôle of energy and the topological Peter-Weyl theorem.The final SELF-CONTAINED version is v.2. The current version had some material removed to the prequels [FHT1,2]. Minor notational changesAlgebraic TopologyHigh Energy Physics - TheoryK-Theory and HomologyRepresentation Theory19L47, 22E67 (primary); 19K99 (secondary)text