In mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring, is a result from operator theory that represents any completely positive map on a C*-algebra as a composition of two completely positive maps each of which has a special form: A *-representation of A on some auxiliary Hilbert space K followed by An operator map of the form T → V*TV. Moreover, Stinespring's theorem is a structure theorem from a C*-algebra into the algebra of bounded operators on a Hilbert space. Completely positive maps are shown to be simple modifications of *-representations, or sometimes called *-homomorphisms. Formulation In the case of a unital C*-algebra, the result is as follows: Theorem. Let A be a unital C*-algebra, H be a Hilbert space, and B(H) be the bounded operators on H. For every completely positive there exists a Hilbert space K and a unital *-homomorphism such that where is a bounded operator. Furthermore, we have Informally, one can say that every completely positive map can be "lifted" up to a map of the form . The converse of the theorem is true trivially. So Stinespring's result classifies completely positive maps. Sketch of proof We now briefly sketch the proof. Let . For , define and extend by semi-linearity to all of K. This is a Hermitian sesquilinear form because is compatible with the * operation. Complete positivity of is then used to show that this sesquilinear form is in fact positive semidefinite. Since positive semidefinite Hermitian sesquilinear forms satisfy the Cauchy–Schwarz inequality, the subset is a subspace. We can remove degeneracy by considering the quotient space . The completion of this quotient space is then a Hilbert space, also denoted by . Next define and . One can check that and have the desired properties. Notice that is just the natural algebraic embedding of H into K. One can verify that holds. In particular holds so that is an isometry if and only if . In this case H can be embedded, in the Hilbert space sense, into K and , acting on K, becomes the projection onto H. Symbolically, we can write In the language of dilation theory, this is to say that is a compression of . It is therefore a corollary of Stinespring's theorem that every unital completely positive map is the compression of some *-homomorphism. Minimality The triple (, V, K) is called