Characters of $C(X)$

Question

I showed that the character space $\Omega (\ell^1 (\mathbb Z))$ is homeomorphic to $S^1$.

Now I am wondering if there is a similar identification for $C(X)$ where $X$ is compact Hausdorff with the sup norm?

What is your definition of 'the character space' of a Banach space? Besides $\ell^1(N)$ is isomorphic to $\ell^1(Z)$. — user10676, Apr 15 '14 at 15:27

score 1 · Accepted Answer · edited Apr 13 '17 at 12:20

1

Yes. The character space of $C(X)$ is homemorphic to $X$.

Proving this is a good exercise, but if you need a hint, the key ideas are contained in this question.

edited Apr 13 '17 at 12:20

Community

answered Apr 18 '14 at 12:12

Nate Eldredge

And is $C(X)$ homeomorphic to the character space of some space? – Student Apr 20 '14 at 07:48
@Student: Thanks to Banach-Alaoglu, the character space of any unital commutative Banach algebra is compact. If not unital, it's still locally compact. But $C(X)S is locally compact only when X is finite. – Nate Eldredge Apr 20 '14 at 11:35
Thank you for your helpful comments! – Student Apr 21 '14 at 07:03

1 Answers1