Semigroup representations, positive definite functions and abelian $C^*$-algebras

It is shown that every $*$-representation of a commutative semigroup $S$ with involution via operators on a Hilbert space has an integral representation with respect to a unique, compactly supported, selfadjoint Radon spectral measure defined on the Borel sets of the character space of $S$. The main feature is that the proof, which is based on the theory of positive definite functions, makes no use what-so-ever (directly or indirectly) of the theory of $C^*$-algebras or more general Banach algebra arguments. Accordingly, this integral representation theorem is used to give a new proof of the Gelfand-Naimark theorem for abelian $C^*$-algebras.