a Stinespring representation of Φ. A natural question is now whether one can reduce a given Stinespring representation in some sense. Let K1 be the closed linear span of (A) VH. By property of *-representations in general, K1 is an invariant subspace of (a) for all a. Also, K1 contains VH. Define We can compute directly and if k and ℓ lie in K1 So (1, V, K1) is also a Stinespring representation of Φ and has the additional property that K1 is the closed linear span of (A) V H. Such a representation is called a minimal Stinespring representation. Uniqueness Let (1, V1, K1) and (2, V2, K2) be two Stinespring representations of a given Φ. Define a partial isometry W : K1 → K2 by On V1H ⊂ K1, this gives the intertwining relation In particular, if both Stinespring representations are minimal, W is unitary. Thus minimal Stinespring representations are unique up to a unitary transformation. Some consequences We mention a few of the results which can be viewed as consequences of Stinespring's theorem. Historically, some of the results below preceded Stinespring's theorem. GNS construction The Gelfand–Naimark–Segal (GNS) construction is as follows. Let H in Stinespring's theorem be 1-dimensional, i.e. the complex numbers. So Φ now is a positive linear functional on A. If we assume Φ is a state, that is, Φ has norm 1, then the isometry is determined by for some of unit norm. So and we have recovered the GNS representation of states. This is one way to see that completely positive maps, rather than merely positive ones, are the true generalizations of positive functionals. A linear positive functional on a C*-algebra is absolutely continuous with respect to another such functional (called a reference functional) if it is zero on any positive element on which the reference positive functional is zero. This leads to a noncommutative generalization of the Radon–Nikodym theorem. The usual density operator of states on the matrix algebras with respect to the standard trace is nothing but the Radon–Nikodym derivative when the reference functional is chosen to be trace. Belavkin introduced the notion of complete absolute continuity of one completely positive map with respect to another (reference) map and proved an operator variant of the noncommutative Radon–Nikodym theorem for completely positive maps. A particular case of this theorem corresponding to a tracial completely positive reference map on the matrix algebras leads