  We present a new approach to noncommutative real algebraic geometry based on
the representation theory of $C^\ast$-algebras. An important result in
commutative real algebraic geometry is Jacobi's representation theorem for
archimedean quadratic modules on commutative rings, \cite[Theorem 5]{jacobi}.
We show that this theorem is a consequence of the Gelfand-Naimark
representation theorem for commutative $C^\ast$-algebras. A noncommutative
version of Gelfand-Naimark theory was studied by I. Fujimoto. We use his
results to generalize Jacobi's theorem to associative rings with involution.
