# Extension of a bounded linear functional

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.

• I don't understand what your question is. The linked paper makes no such claim, and is very careful to point out that you need extra hypotheses to extend functionals. Jul 1, 2022 at 13:21
– Piku
Jul 1, 2022 at 15:41
• You have now edited away the reference to where "it was concluded that" there exists an extension. Jul 1, 2022 at 18:09
• @Piku: The point is, it is unclear right now if you are saying "I read this specific fact asserted somewhere and I want to know why it's true" or if you are saying "I am wondering if the following might be true." Jul 1, 2022 at 19:01
• @Piku: in that case, it would be really helpful for everyone trying to answer your question if you cited specifically where you read it. Jul 1, 2022 at 19:20

• The paper considers only positive maps $$\phi:A\rightarrow M_n$$;
• The paper does not consider $$A \widehat\otimes M_n$$ but rather $$A \widehat\otimes T_n$$ where $$T_n$$ is the $$n\times n$$ matrices with the trace class norm.
The paper (or really an "addenda") is hard to read, because you need access to Stormer's book, reference [6]. However, this all said, I agree with you that the argument seems wrong. The only thing [6, Lemma 4.2.2] shows is that the dual of $$A \widehat\otimes T_n$$ is the bounded linear maps $$A\rightarrow M_n$$. Of course, this says nothing about the inclusion
$$A \widehat\otimes T_n \rightarrow B(H) \widehat\otimes T_n$$