Extension of the projective norm to a cross norm Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ be an operator system. Is it possible to extend the projective norm (the greatest cross norm) of $\mathcal{A}\otimes M_n$ to a cross norm on $\mathcal{B}(\mathcal{H})\otimes M_n$?
Any comment is highly appreciated. Please suggest to me some reference in case it exists in the literature.
 A: By request, some expansion of my easy comment.  Given normed spaces $E,F$ and $\newcommand{\mc}{\mathcal}u\in E\otimes F$ write $\pi(u; E\otimes F)$ for the projective norm of $u$ in $E\otimes F$.
The question asks if there is a crossnorm $\|\cdot\|$ on $\mc B(\mc H)\otimes M_n$ with
$$ \|u\| = \pi(u; \mc A\otimes M_n) \qquad (u\in\mc A\otimes M_n). $$
However, we know two things:

*

*$\|u \| \leq \pi(u; \mc B(\mc H)\otimes M_n)$ for any $u\in\mc B(\mc H)\otimes M_n$ because $\pi$ is the greatest crossnorm;

*$\pi(u; \mc B(\mc H)\otimes M_n) \leq \pi(u; \mc A\otimes M_n)$ by definition of the projective norm.

Putting these together gives
$$ \pi(u; \mc A\otimes M_n) = \|u\| \leq \pi(u; \mc B(\mc H)\otimes M_n)
\leq \pi(u; \mc A\otimes M_n) $$
for each $u\in\mc A\otimes M_n$.  So we have equality throughout, meaning:

*

*$\|\cdot\| = \pi(\cdot; \mc B(\mc H)\otimes M_n)$ on $\mc A\otimes M_n$;

*So the original question has a positive answer exactly when $\pi(u; \mc B(\mc H)\otimes M_n) = \pi(u; \mc A\otimes M_n)$ for all $u$.

That is, maybe there is some choice in $\|\cdot\|$ but that choice is unimportant, because if any crossnorm will work, the projective norm will work.
