Kantorovich's theorem (a version of Hahn Banach's theorem for positive linear functionals)

Question

I am having some trouble to complete the proof of the following theorem:

Consider the set of continuous functions from a compact topological space $X$ into $\mathbb{R}$, namely $C(X,\mathbb{R})$, equipped with the supremum norm. And let $\varphi:C(X,\mathbb{R})\longrightarrow\mathbb{R}$ a positive linear functional. Then there exists a positive linear functional $\overline{\varphi}:L^{\infty}(X)\longrightarrow\mathbb{R}$ such that $\overline{\varphi}\mid_{C(X,\mathbb{R})}=\varphi$ and $\parallel\overline{\varphi}$$\parallel_{\infty}=\parallel$$\varphi\parallel_{\infty}$.

According to the book where I found this theorem, it follows easily from the standard version of Hahn-Banach's theorem, and for the most part it seems to be the case: the authors define a sublinear function on $L^{\infty}(X)$ that agrees with $\varphi$ on $C(X,\mathbb{R})$, proceed to apply Hahn-Banach's theorem and in fact show that this extension is also positive. Now this is the problem. The version of Hahn-Banach's theorem used in this proof doesn't guarantee that the extended functional $\overline{\varphi}$ has the same norm as $\varphi$. It doesn't even guarantee that the extended functional is bounded. The authors don't address this issue at all, as if it was part of the conclusion of Hahn-Banach's theorem. I am aware that there is a way to extend any linear functional without increasing it's norm but the theorems at hand do not guarantee that the resulting extension is positive.

I think that this other theorem should be enough to ensure that the extended functional is bounded and doesn't increase the norm:

Let $X$ be a topological space and let $\varphi$ be a positive linear functional on $L^{\infty}(X)$. Then $\varphi$ is bounded with $\parallel\varphi\parallel_{\infty}$=$\varphi(1)$.

So, the theorem above ensures boundedness of the extended functional and also since $\overline{\varphi}(1)=\varphi(1)$ the two norms should be equal. Is this line of reasoning correct?

Answer 1

The result you quote is exactly what you need (and it holds in way more generality, for any linear functional on a C$^*$-algebra, even on an operator system). Properly, what you need is the converse, which also holds.

In your case, $1\in C(X,\mathbb R)$ and so by the result you quote you have $\|\varphi\|=\varphi(1)$. Now extend by Hahn-Banach, and you get $\tilde\varphi$ with $\|\tilde\varphi\|=\|\varphi\|=\varphi(1)=\tilde\varphi(1)$. So the converse of the quoted result gives you that $\tilde\varphi$ is positive.