Weak * Compactness and Local Compactness

Question

I'm reading through Murphy's C* Algebras, and the following theorem is presented:

If $A$ is an abelian Banach algebra, then the set of characters $\Omega(A)$ is a locally compact Hausdorff space. If $A$ is unital, then $\Omega(A)$ is compact.

Proof: It's easily checked that $\Omega(A) \cup \{0\}$ is weak* closed in the weak * compact closed unit ball $S$ of $A^*$. Thus, $\Omega(A) \cup \{0\}$ is weak * compact implying that $\Omega(A)$ is locally compact. If $A$ is unital, then $\Omega(A)$ is weak * closed in $S$, and hence $\Omega(A)$ is compact.

So I have two main questions here.

1. Why does weak * compactness of $\Omega(A) \cup \{0\}$ imply local compactness of $\Omega(A)$?
2. Does the term "compact" is the last sentence of the proof mean weak* compact? If not, what does it mean simply by compact?

I'm brand new to this subject, so any help would be really appreciated.

Answer 1

(1) Note that $\Omega(A)$ is open in $\Omega(A)\cup\{0\}$. Then invoke the general fact that open subspaces of locally compact spaces are again locally compact.

(2) Yes, compact here means compact in the weak$^*$-topology.