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}