Added the unitisation.

src/op/banach/definitions.tex

@@ -27,5 +27,44 @@
     \end{enumerate}
 \end{definition}
 
+\begin{definition}[Unital Homomorphism]
+\label{definition:banach-algebra-unital-homomorphism}
+    Let $A, B$ be unital Banach algebras and $\phi: A \to B$ be a homomorphism, then $\phi$ is a \textbf{unital homomorphism} if $\phi(1) = 1$. 
+\end{definition}
+
+\begin{definition}[Unitisation]
+\label{definition:unitisation}
+    Let $A$ be a Banach algebra over $\complex$, and $\tilde A = \complex \oplus A$ with
+    \[
+    \iota: A \to \complex \oplus A \quad x \mapsto 0 + x
+    \]
+    
+    For each $\lambda + x, \mu + y \in \tilde A$, define
+    \[
+    (\lambda + x)(\mu + y) = \lambda \mu + (\lambda x + \mu x + xy)
+    \]
+    
+    and
+    \[
+    \norm{\lambda + x}_{\tilde A} = |\lambda| \norm{x}_A
+    \]
+
+    then
+    \begin{enumerate}
+        \item $\tilde A$ is a unital associative algebra over $\complex$.
+        \item $\iota: A \to \tilde A$ is a homomorphism. 
+        \item[(U)] For any pair $(B, \phi)$ satisfying (1) and (2), there exists a unique continuous unital homomorphism $\tilde \phi: \tilde A \to B$ such that $\phi(1) = 1$ and the following diagram commutes:
+        \xymatrix{
+        \tilde A \ar@{->}[r]^{\tilde \phi } & B \\
+        A \ar@{->}[u]^{\iota} \ar@{->}[ru]_{\phi} & 
+        }
+        \item $\iota(A)$ is a closed two-sided ideal of $\tilde A$. 
+    \end{enumerate}
+
+    The algebra $\tilde A$ is the \textbf{unitisation} of $A$. 
+\end{definition}
+\begin{proof}
+    (U): For each $\lambda + x \in \tilde A$, let $\tilde \phi(\lambda + x) = \lamdba + \phi(x)$. 
+\end{proof}
 
 

src/op/banach/gelfand.tex

@@ -19,7 +19,6 @@
 \label{proposition:gelfand-transform-gymnastics}
     Let $A$ be a commutative unital Banach algebra and $x \in A$, then:
     \begin{enumerate}
-        \item $\Gamma_A$ is a contractive homomorphism. 
         \item $\Gamma_A(1) = 1$. 
         \item $x \in G(A)$ if and only if $\Gamma_A x \in G(C(\Omega(A); \complex))$. 
         \item $(\Gamma_Ax)(\Omega(A)) = \sigma_A(x)$. 

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}