Exercise 4, § 3

Let A and B be star algebras. A linear mapping $u$ of A into B is said to be positive if $u(A_+)\subset B_+$, and totally positive if, for every integer $k\in \mathbf{N}$, the mapping $u_k= 1_{\mathbf{C}^k}\otimes u:\mathbf{C}^k\otimes A\rightarrow \mathbf{C}^k\otimes B$ is positive.

a) Let A be a unital star algebra, E and F complex Hilbert spaces, $v$ a linear mapping of E into F and $\varphi$ a unital morphism of involutive algebras of A into $\mathscr{L}(F)$ such that $\|\varphi \|^2=\|\varphi (1)\|$. The mapping $u: A\rightarrow \mathscr{L}(E)$ defined by $u(a) =v^*\circ \varphi (a)\circ v$ is totally positive.

The aim of the exercise is to prove the converse of this assertion («Stinespring theorem»). Let A be a unital star algebra, E a complex Hilbert space, and $u: A\rightarrow \mathscr{L}(E)$ a totally positive linear mapping.

b) Let $F = A\otimes E$. There exists a positive sesquilinear form $q$ on F such that

for all $(a, b)\in A^2$ and $(x, y)\in E^2$.

Let $\widehat{F}$ be the separated-completed Hilbert space of F for the sesquilinear form $q$.

c) Let $b\in A$. The endomorphism of F such that $a\otimes x\mapsto ba\otimes x$ for all $a\in A$ and $x\in E$ induces by passing to the quotients an endomorphism $\varphi (b)$ of $\widehat{F}$ such that $\|\varphi (b)\|\leqslant \|b\|$; the mapping $\varphi : A\rightarrow \mathscr{L}(F)$ is a unital morphism of involutive algebras.

d) Let $v$ be the linear mapping of E into $\widehat{F}$ such that $v(x)$ is the class of $1\otimes x$ for all $x\in E$. We have $\|v\|^2=\|\varphi (1)\|$, and $u(a) =v^*\circ \varphi (a)\circ v$ for all $a\in A$.