Definition 33.7.4 (Space of Multiplicative Linear Functionals).label Let $A$ be a Banach algebra, then $\Omega(A)$ is the space of multiplicative linear functionals, and with respect to the weak-* topology,
- (1)
If $A$ is unital, then $\Omega(A)$ is a compact Hausdorff space.
- (2)
$\Omega(A) \cup \bracs{0}$ is a compact Hausdorff space, and $\Omega(A)$ is an LCH space.
Proof. (1): By Proposition 33.7.2, $\Omega(A)$ is a weak-* closed subset of $\bracsn{\phi \in A^*:\norm{\phi}_{A^*} = 1}$, so it is compact by the Banach-Alaoglu Theorem.
(2): By Proposition 33.7.3, $\Omega(A) \cup \bracs{0}$ is a weak-* closed subset of $\ol{B_{A^*}(0, 1)}$, so it is compact by the Banach-Alaoglu Theorem.$\square$
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