Theorem: Spectrum of Banach Algebra is Weak-* Locally Compact Hausdorff Space

Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $\struct {\Phi_A, w^\ast}$ be the spectrum of $A$. 
Then $\struct {\Phi_A, w^\ast}$ is a locally compact Hausdorff space.

Proof:
From Weak-* Topology is Hausdorff, $\struct {\Phi_A, w^\ast}$ is a Hausdorff space.
By the definition of a character, we have:
:$\Phi_A = \set {\phi \in A^\ast : \phi \ne {\mathbf 0}_{A^\ast}, \, \map \phi x \map \phi y = \map \phi {x y} \text { for all } x, y \in A}$
hence:
:$\Phi_A \cup \set { {\mathbf 0}_{A^\ast} } = \set {\phi \in A^\ast : \map \phi x \map \phi y = \map \phi {x y} \text { for all } x, y \in A}$
From Character on Banach Algebra is Continuous, we have:
:$\norm \phi_{A^\ast} \le 1$
Hence:
:$\Phi_A \cup \set { {\mathbf 0}_{A^\ast} } = \set {\phi \in B_{A^\ast} : \map \phi x \map \phi y = \map \phi {x y} \text { for all } x, y \in A}$
where $B_{A^\ast}$ is the closed unit ball of $\struct {A^\ast, \norm {\, \cdot \,}_{A^\ast} }$.
For each $x \in A$, define:
:$\map {x^\wedge} \phi = \map \phi x$
for each $\phi \in A^\ast$.
From:
:Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual
and:
:Characterization of Continuity of Linear Functional in Weak-* Topology
we have:
:$x^\wedge : \struct {A^\ast, w^\ast} \to \C$ is continuous for each $x \in A$.
Write:
{{begin-eqn}}
{{eqn	| l = \Phi_A \cup \set { {\mathbf 0}_{A^\ast} }
		| r = \bigcap_{x, y \in A} \set {\phi \in B_{A^\ast} : \map {x^\wedge} \phi \map {y^\wedge} \phi - \map {\paren {x y}^\wedge} \phi = 0}
}}
{{eqn	| r = \bigcap_{x, y \in A} B_{A^\ast} \cap \paren {x^\wedge y^\wedge - \paren {x y}^\wedge}^{-1} \sqbrk {\set 0}
}}
{{end-eqn}}
From Sum of Continuous Functions on Topological Ring is Continuous, we have:
:$x^\wedge y^\wedge - \paren {x y}^\wedge : \struct {A^\ast, w^\ast} \to \C$ is continuous for each $x, y \in A$.
From the definition of a continuous mapping, we have:
:$\paren {x^\wedge y^\wedge - \paren {x y}^\wedge}^{-1} \sqbrk {\set 0}$ is closed in $\struct {A^\ast, w^\ast}$.
From the Banach-Alaoglu Theorem, we have:
:$\struct {B_{A^\ast}, w^\ast}$ is compact.
From Compact Subspace of Hausdorff Space is Closed:
:$B_{A^\ast}$ is closed in $\struct {A^\ast, w^\ast}$.
From Intersection of Closed Sets is Closed: Topology:
:$B_{A^\ast} \cap \paren {x^\wedge y^\wedge - \paren {x y}^\wedge}^{-1} \sqbrk {\set 0}$ is closed in $\struct {A^\ast, w^\ast}$.
Again from Intersection of Closed Sets is Closed: Topology, we have:
:$\ds \bigcap_{x, y \in A} B_{A^\ast} \cap \paren {x^\wedge y^\wedge - \paren {x y}^\wedge}^{-1} \sqbrk {\set 0}$ is closed in $\struct {A^\ast, w^\ast}$.
From Intersection is Subset, we have:
:$\ds \Phi_A \cup \set { {\mathbf 0}_{A^\ast} } = \bigcap_{x, y \in A} B_{A^\ast} \cap \paren {x^\wedge y^\wedge - \paren {x y}^\wedge}^{-1} \sqbrk {\set 0} \subseteq B_{A^\ast}$
From the Banach-Alaoglu Theorem:
:$\struct {B_{A^\ast}, w^\ast}$ is compact.
From Closed Subspace of Compact Space is Compact:
:$\struct {\Phi_A \cup \set { {\mathbf 0}_{A^\ast} }, w^\ast}$ is compact.
From $T_2$ (Hausdorff) Space is $T_1$ Space, we have:
:$\struct {A^\ast, w^\ast}$ is a $T_1$ Space. 
Hence:
:$\set { {\mathbf 0}_{A^\ast} }$ is closed in $\struct {A^\ast, w^\ast}$.
Hence:
:$\Phi_A = \paren {\Phi_A \cup \set { {\mathbf 0}_{A^\ast} } } \setminus \set { {\mathbf 0}_{A^\ast} }$ is open in $\struct {\Phi_A \cup \set { {\mathbf 0}_{A^\ast} }, w^\ast}$.
From Open Subset of Locally Compact Hausdorff Space is Locally Compact Hausdorff Space, we have:
:$\struct {\Phi_A, w^\ast}$ is a locally compact Hausdorff space.
{{qed}}
Category:Spectrum of Banach Algebra is Weak-* Locally Compact Hausdorff Space
Category:Locally Compact Hausdorff Spaces
Category:Weak-* Topologies
Category:Spectra (Banach Algebras)
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