Added Gelfand Naimark.

src/op/c-star/homomorphism.tex

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+\section{*-Homomorphisms}
+\label{section:c-star-homomorphism}
+
+\begin{definition}[*-Homomorphism]
+\label{definition:c-star-homomorphism}
+    Let $A, B$ be involutive algebras over $\complex$ and $\phi: A \to B$, then $\phi$ is a \textbf{*-homomorphism} if:
+    \begin{enumerate}[label=(SH\arabic*)]
+        \item For each $x, y \in A$ and $\lambda \in \complex$, $\phi(\lambda x + y) = \lambda \phi(x) + \phi(y)$. 
+        \item For each $x, y \in A$, $\phi(xy) = \phi(x)\phi(y)$. 
+        \item For each $x \in A$, $\phi(x^*) = \phi(x)^*$.
+    \end{enumerate}
+
+    If $A$ and $B$ are unital, then $\phi$ is \textbf{unital} if:
+    \begin{enumerate}
+        \item[(U)] $\phi(1_A) = \one_B$. 
+    \end{enumerate}
+\end{definition}
+
+
+\begin{proposition}
+\label{proposition:star-homomorphism-contractive}
+    Let $A, B$ be unital $C^*$-algebras and $\phi: A \to B$ be a unital *-homomorphism, then for each $x \in A$, 
+    \begin{enumerate}
+        \item $\sigma_B(\phi(x)) \subset \sigma_A(x)$. 
+        \item $\norm{\phi(x)}_B \le \norm{x}_A$. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1): Since $\phi$ is unital, $\phi(G(A)) \subset G(B)$, so $\sigma_B(\phi(x)) \subset \sigma_A(x)$. 
+    
+    (2): By (1) and \autoref{corollary:c-star-unique-norm}, 
+    \begin{align*}
+    \norm{\phi(x)}_B^2 &= \sup\bracsn{|\lambda|\ | \lambda \in \sigma_B(\phi(x^*x))} \\
+    &\ge \sup\bracsn{|\lambda|\ | \lambda \in \sigma_A(x^*x)} = \norm{x}_A^2
+    \end{align*}
+    
+\end{proof}
+
+