  In this paper we develop a method of constructing Hilbert spaces and the
representation of the formal algebra of quantum observables in deformation
quantization which is an analog of the well-known GNS construction for complex
$C^*$-algebras: in this approach the corresponding positive linear functionals
(`states') take their values not in the field of complex numbers, but in (a
suitable extension field of) the field of formal complex Laurent series in the
formal parameter. By using the algebraic and topological properties of these
fields we prove that this construction makes sense and show in physical
examples that standard representations as the Bargmann and Schr\"odinger
representation come out correctly, both formally and in a suitable convergence
scheme. For certain Hamiltonian functions (contained in the Gel'fand ideal of
the positive functional) a formal solution to the time-dependent Schr\"odinger
equation is shown to exist. Moreover, we show that for every K\"ahler manifold
equipped with the Fedosov star product of Wick type all the classical delta
functionals are positive and give rise to some formal Bargmann representation
of the deformed algebra.
