Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I have the following question,

Is it possible to get somehow a compact Hausdorff space $X$ which is second-countable from a unital commutative C*-algebra. If it is possible, what should we assume for our C*-algebra. Gelfand-Naimark theorem gives us $C(X)$, where $X$ is a compact Hausdorff space, but I'm asking how to get it with the second axiom of countability. Thank you in advance for any answers or hints.

share|improve this question
add comment

1 Answer

up vote 4 down vote accepted

The space $X$ is second countable if and only if $C(X)$ is separable for the norm. This is proved, for example, as Theorem 2.4 of the little article "Notes on the Separability of C* algebras" by Chun-Yen Chou. Actually, the short proof given there works also for locally compact Hausdorff spaces and therefore non-unital $C^{*}$-algebras.

share|improve this answer
1  
It is also a standard exercise in text books. –  Bill Johnson Aug 30 '12 at 0:31
    
You're right Bill. This was probably best left as a long comment, should the OP have liked to move the question to math.stackexchange. –  Jon Bannon Aug 30 '12 at 0:54
add comment

Your Answer

 
discard

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.