Matrix order unit norm

Let $L$ be an operator system. We define norms by

$$\Vert x\Vert _n:=\inf\left\{r\in{\mathbb{R}}\vert \begin{pmatrix}... ...!\:l}_n & x\\ x^* & r\mathrm{1\!\!\!\:l}_n\end{pmatrix}\in M_{2n}(L)_+\right\}$$ (1)

for all $n\in{\mathbb{N}}$ and $x\in M_n(L)$. With these norms $L$ becomes an operator space.

If $\Phi$ is any unital completely positve embedding from $L$ into some $B(\H)$ (cf. section ) then $\Vert\Phi^{(n)}(x)\Vert=\Vert x\Vert _n$ for all $n\in{\mathbb{N}}$ and $x\in M_n(L)$. This holds because $\Vert y\Vert\leq 1$ if and only if

$$0\leq\begin{pmatrix}\mathrm{1\!\!\!\:l} & y\\ y^* & \mathrm{1\!\!\!\:l} \end{pmatrix}$$

for all $y\in B(H)$ and all Hilbert spaces $\H$.

Let $L$ and $S$ be operator systems and let $\psi:L\rightarrow S$ be completely positive. We supply $L$ and $S$ with the norms from equation (). Then $\psi$ is completely bounded and $\Vert\psi(\mathrm{1\!\!\!\:l})\Vert=\Vert\psi\Vert=\Vert\psi\Vert _{cb}$ (cf. [Pau86, Proposition 3.5]).