Stinespring factorization theorem

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:

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.

In the case of a unital C*-algebra, the result is as follows:

Informally, one can say that every completely positive map ${\displaystyle \Phi }$ can be "lifted" up to a map of the form ${\displaystyle V(\cdot )V^{*}}$.

The converse of the theorem is true trivially. So Stinespring's result classifies completely positive maps.

We now briefly sketch the proof. Let ${\displaystyle K=A\otimes H}$. For ${\displaystyle a\otimes h,b\otimes g\in K}$, define

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