Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B(\mathcal{H})}$ be an operator system. Suppose $T_n$ is a collection of all $n$-by-$n$ matrices equipped with the trace class norm. Let $\varphi:\mathcal{A}\widehat{\otimes} T_n\rightarrow\mathbb{C}$ be a linear functional with $\Vert\varphi\Vert=1$. Then it was concluded that, by the Hahn-Banach theorem, there exists an extension $\hat{\varphi}:\mathcal{B{\mathcal{(H)}}}\widehat{\otimes} T_n\rightarrow\mathbb{C}$ of $\varphi$ such that $\Vert\hat{\varphi}\Vert=1$. My question is how does this conclusion follow?

This doubt comes while reading Theorem 4 of the paper Extension of positive maps (see page 3, line 9).

It is to be noted that the conclusion would follow easily, by the Hahn-Banach theorem, if the inclusion $\mathcal{A}\widehat{\otimes} T_n\rightarrow\mathcal{B}\mathcal{(H)}\widehat{\otimes} T_n$ is isometric, that is, if the projective norm on $\mathcal{B}\mathcal{(H)}\widehat{\otimes} T_n$ is an extension of the projective norm of $\mathcal{A}\widehat{\otimes} T_n$. But, in general, the inclusion $\mathcal{A}\widehat{\otimes} T_n\rightarrow \mathcal{B\mathcal{(H)}}\widehat{\otimes} T_n$ is norm decreasing. The book "A theory of cross space" by Schatten contains an example (Corollary 3.5, page 57) that the above inclusion is not isometric in general.

However, I think the author means the inclusion $\mathcal{A}\widehat{\otimes} T_n\rightarrow\mathcal{B}\mathcal{(H)}\widehat{\otimes} T_n$ is isometric in this special situation, that is, when $\mathcal{A}$ is an operator system and $\mathcal{H}$ is finite-dimensional. If so, it will lead to the conclusion written in the first paragraph immediately using the Hahn-Banach theorem.

I tried to prove this in various ways but could not able to prove it.

Any comment is highly appreciated. Thanks in advance.

where"it was concluded that" there exists an extension. $\endgroup$3more comments