Stinespring Factorization Theorem - Sketch of Proof

Sketch of Proof

We now briefly sketch the proof. Let . For, define

and extend by linearity to all of K. We see that this is a bilinear form by definition. By the completely positivity of Φ, it is also positive. The assumption that Φ preserves positivity means Φ commutes with the * operation in A, which can be used to show that is conjugate-symmetric. Therefore is a, possibly degenerate, Hermitian bilinear form. Since Hermitian bilinear forms satisfy the Cauchy Schwarz inequality, the subset

is a subspace. We can remove degeneracy by considering the quotient space K / K' . The completion of this quotient space is then a Hilbert space, also denoted by K. Next define and, where 1 is the unit in A. One can check that π and V have the desired properties.

Notice that is just the natural algebraic embedding of H into K. Direct calculation shows that, in the finite dimensional case, can be identified with the algebraic identity map on H. The definitions of and K are also rather natural. Thus the key element of the proof is the introduction of . In particular, after the algebraic embedding, H is "re-normed" in the following sense: If h is identified with, then

This can be viewed as the restriction of to H.

When Φ is unital, i.e., we see that is an isometry and H can be embedded, in the Hilbert space sense, into K. V, acting on K, becomes the projection onto H. Symbolically, we can write

In the language of dilation theory, this is to say that Φ(a) is a compression of π(a). It is therefore a corollary of Stinespring's theorem that every unital completely positive map is the compression of some *-homomorphism.