Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm trying to convince myself that the state space $S(A)$ of a unital $C^*$-algebra is weak* compact. I've proven that $S(A)$ is convex, and I feel that this should allow me to conclude weak* compactness. However, after awhile, I don't see how this could be the case. Can someone point me in the right direction?

share|cite|improve this question
up vote 4 down vote accepted

Use the Banach–Alaoglu theorem. Now you only need to prove that $S(A)$ is weakly* closed.

Elaboration: The weak* topology is, by definition, the weakest topology on $A^*$ for which every bounded linear functional of the form $\psi\mapsto\psi(a)$, with $a\in A$, is continuous. In particular, applying this to the unit element $e$, we conclude that $\{\,\psi\in A^*\colon \psi(e)=1\,\}$ is weakly* closed. The state space is just the intersection of this hyperplane with the unit ball of $A^*$, which is also weakly* closed (which follows from compactness, but more easily from Hahn–Banach).

share|cite|improve this answer
Thanks, Harald. I thought about that, but I do not easily see why $S(A)$ is weak* closed. It seems obvious to everybody but me. Could you elaborate a bit on it? I'm having a hard time understanding what is a weak* closed set. – ragrigg Feb 22 '13 at 8:18
@ragrigg I added an elaboration on that point. – Harald Hanche-Olsen Feb 22 '13 at 8:41
Thanks a lot. I was missing the obvious: the first sentence in your elaboration. By the way, what goes wrong when you don't have a unit? I'm thinking in the non-unital $C^*$-algebra of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ that vanish at infinity. I clearly see that your argument would not work anymore, but I don't see why we will have that $S(A)$ is not compact. Any insight on that one? – ragrigg Feb 22 '13 at 9:53
@ragrigg In the example you mention, just take point evaluation at $n$ and let $n\to\infty$. You get a sequence of states converging weakly* to the zero functional. What happens in a general non-unital C*-algebra, I am not so sure. – Harald Hanche-Olsen Feb 22 '13 at 15:47

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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