  A host algebra of a topological group G is a C^*-algebra whose
representations are in one-to-one correspondence with certain continuous
unitary representations of G. In this paper we present an approach to host
algebras for infinite dimensional Lie groups which is based on complex
involutive semigroups. Any locally bounded absolute value \alpha on such a
semigroup S leads in a natural way to a C^*-algebra C^*(S,\alpha), and we
describe a setting which permits us to conclude that this C^*-algebra is a host
algebra for a Lie group G. We further explain how to attach to any such host
algebra an invariant weak-*-closed convex set in the dual of the Lie algebra of
G enjoying certain nice convex geometric properties. If G is the additive group
of a locally convex space, we describe all host algebras arising this way. The
general non-commutative case is left for the future.
