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?

  • 1
    $\begingroup$ 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. $\endgroup$ – Giuseppe Negro May 13 '14 at 12:21

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

  • $\begingroup$ 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 $n$. $\endgroup$ – Yan kai May 13 '14 at 18:01
  • $\begingroup$ Exactly :) $ \ $ $\endgroup$ – Martin Argerami May 13 '14 at 18:18
  • $\begingroup$ Oh, thanks Martin. :P $\endgroup$ – Yan kai May 13 '14 at 18:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.