# An application of Hahn-Banach (separation) theorem

Here is a quotation of a book:

Let $S(A)$ denote the state space of a C*-algebra $A$ and $M\subset S(A)$ denote a weak-$*$ closed convex set. Assume there is a state $\psi$ which does not belong to $M$. By the Hahn-Banach Theorem we can find $a\in A$ and a real number $t$ such that $$Re(\phi(a))<t<Re\psi(a).$$ for every $\phi\in M.$

In my view, if $M$ is a closed convex set, then, from Hahn-Banach Theorem, we have a continuous linear functional $\rho$ on $M$ such that $Re(\rho(\phi))<t\leq Re(\rho(\psi))$. But, how to find such a $a\in A$ in the quotation?

• The idea has surely to do with the fact that the elements of $A$ are weak-* continuous linear functionals on $M$, by construction of the weak-* topology. What I don't know is if all functionals are elements of $A$. I guess so, but I am not sure. – Giuseppe Negro May 13 '14 at 12:21

## 1 Answer

Your $\rho$ is not in $M$ but in $A^{**}$; and $A$ is dense in $A^{**}$.

• Oh, you mean: $A$ is weak-$*$ dense in $A^{**}$, so we can find $\{a_{n}\}$ such that $a_{n}(\phi) \rightarrow \rho(\phi)$, for any $\phi \in A^{*}$. This implies $Re a_{n}(\phi) \rightarrow Re \rho (\phi)$. Hence, we can find $a_{n}$ for a large enough $n$. – Yan kai May 13 '14 at 18:01
• Exactly :) $\$ – Martin Argerami May 13 '14 at 18:18
• Oh, thanks Martin. :P – Yan kai May 13 '14 at 18:27