Added the unitisation.

src/op/banach/multiplicative.tex

@@ -14,7 +14,6 @@
         \item $\norm{\phi}_{A^*} = 1$. 
         \item $\phi(G(A)) \subset \complex \setminus \bracs{0}$. 
     \end{enumerate}
-    
 \end{proposition}
 \begin{proof}
     (3): For each $x \in G(A)$, $1 = \phi(xx^{-1}) = \phi(x)\phi(x^{-1}) \ne 0$. 
@@ -24,12 +23,27 @@
     (2): For each $\lambda \in \complex$, $\phi(\lambda 1) = \lambda$, so $\norm{\phi}_{A^*} \le 1$. 
 \end{proof}
 
+\begin{proposition}
+\label{proposition:multiplicative-less-unit}
+    Let $A$ be a Banach algebra and $\phi \in A^*$ be a multiplicative functional, then $\norm{\phi}_{A^*} \le 1$. 
+\end{proposition}
+\begin{proof}
+    Let $\tilde A$ be the unitisation of $A$, then by (U) of the \hyperref[unitisation]{definition:unitisation}, $\phi$ extends to a multiplicative functional $\tilde \phi$ on $\tilde A$. Therefore $\norm{\phi}_{A^*} \le \normn{\tilde \phi}_{{\tilde A}^*} = 1$. 
+\end{proof}
+
+
 \begin{definition}[Space of Multiplicative Linear Functionals]
 \label{definition:multiplicative-linear-functional-space}
-    Let $A$ be a unital Banach algebra, then $\Omega(A)$ is the \textbf{space of multiplicative linear functionals}, which is a compact Hausdorff space under the weak-* topology. 
+    Let $A$ be a Banach algebra, then $\Omega(A)$ is the \textbf{space of multiplicative linear functionals}, and with respect to the weak-* topology, 
+    \begin{enumerate}
+        \item If $A$ is unital, then $\Omega(A)$ is a compact Hausdorff space. 
+        \item $\Omega(A) \cup \bracs{0}$ is a compact Hausdorff space, and $\Omega(A)$ is a LCH space.
+    \end{enumerate}
 \end{definition}
 \begin{proof}
-    By the \hyperref[Banach-Alaoglu Theorem]{theorem:alaoglu}.
+    (1): By \autoref{proposition:multiplicative-unit}, $\Omega(A)$ is a weak-* closed subset of $\bracsn{\phi \in A^*:\norm{\phi}_{A^*} = 1}$, so it is compact by the \hyperref[Banach-Alaoglu Theorem]{theorem:alaoglu}.
+
+    (2): By \autoref{proposition:multiplicative-less-unit}, $\Omega(A) \cup \bracs{0}$ is a weak-* closed subset of $\ol{B_{A^*}(0, 1)}$, so it is compact by the \hyperref[Banach-Alaoglu Theorem]{theorem:alaoglu}.
 \end{proof}
 
 \begin{proposition}