Stinespring dilation theorem - Imported Knowlpedia Content

Statement

Let $A$ be a unital $C^*$ -algebra, $H$ a Hilbert space, and $\Phi:A\to B(H)$ a completely positive map. The Stinespring dilation theorem provides a Hilbert space $K$ , a unital $*$ -representation $\pi:A\to B(K)$ , and a bounded operator $V:H\to K$ such that

\Phi(a)=V^*\pi(a)V\qquad(a\in A).

One may require minimality, $K=\overline{\pi(A)VH}$ ; then the triple $(K,\pi,V)$ is unique up to a unique intertwining unitary. Conversely, every map of the displayed form is completely positive.

Construction

On the algebraic tensor product $A\odot H$ , define

\left\langle\sum_i a_i\otimes\xi_i,\sum_j b_j\otimes\eta_j\right\rangle =\sum_{i,j}\langle\xi_i,\Phi(a_i^*b_j)\eta_j\rangle.

Complete positivity makes this form nonnegative. Quotienting its null space and completing gives $K$ ; left multiplication gives $\pi$ , and $V\xi$ is represented by $1_A\otimes\xi$ . This is Stinespring's original construction Stinespring, Theorem 1.

Norm and unitality consequences

The construction yields

\lVert\Phi\rVert=\lVert\Phi(1_A)\rVert=\lVert V\rVert^2.

If $\Phi$ is unital completely positive, then $V^*V=1_H$ , so $V$ is an isometry and $\Phi$ is a compression of $\pi$ . A $*$ -homomorphism is the special case in which the compression is unnecessary.

Conventions and scope

For a nonunital $A$ , an analogous theorem uses a nondegenerate representation after passing through an approximate identity or a suitable unitization. For completely positive maps $A\to B$ with abstract $C^*$ -algebra codomain, representing $B$ faithfully on a Hilbert space produces the operator-valued form above; Hilbert-module versions require a separate module formulation.

References

W. Forrest Stinespring, “Positive Functions on $C^*$ -Algebras,” Proceedings of the American Mathematical Society 6 (1955), 211–216. DOI record. Relevant: Theorem 1 and the representation construction.
Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002. DOI record. Relevant: Chapter 4 on Stinespring dilation, minimality, uniqueness, and norm consequences.