I just started studying operators space and systems. I stumbled upon this Lemma 8.1 of the book "Completely bounded maps and operator algebras" by Vern I. Paulsen (1951-):
LEMMA 8.1:
Let $\mathcal{A}$, $\mathcal{B}$ be C^*algebra with unit 1, let $\mathcal{M}$ be an operator space in A, and let $\phi: \mathcal{M} \to \mathcal{B} $. Define an operator system $\mathcal{S}_{\mathcal{M}} \subset M_2(\mathcal{A})$ by $$ \mathcal{S}_{\mathcal{M}}= \{ \begin{bmatrix}\lambda1&a\\b^*&\mu1\end{bmatrix}: \lambda, \mu \in \mathbb{C}, a, b, \in \mathcal{M} \}$$, and $\Phi:\mathcal{S}_{\mathcal{M}}\to M_2(\mathcal{B})$ via $$\Phi\begin{bmatrix}\lambda1&a\\b^*&\mu1\end{bmatrix}=\begin{bmatrix}\lambda1&\phi(a)\\\phi(b)^*&\mu1\end{bmatrix}.$$ If $\phi$ is completelty contractive, then $\Phi$ is completely positive.
I was wondering if the other implication is also true; I guess not otherwise the author would have written "if and if", instead that just "if", but I can find a counterexample.
Does someone know something more?
Thanks in advance
The converse is also true, but considerably easier to prove.
Note that $\Phi$ is ucp, thus it is also completely contractive. Since the maps $$\iota: \mathcal{M}\to S_\mathcal{M}: a \mapsto \begin{pmatrix}0 & a\\0 & 0\end{pmatrix}$$ $$\pi: M_2(\mathcal{B})\to \mathcal{B}: \begin{pmatrix}s & t \\u & v\end{pmatrix}\mapsto t$$ are completely contractive, the composition $$\phi = \pi\circ \Phi \circ \iota$$ is completely contractive.