\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)] $\phi(1_A) = 1_B$. 
    \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}

\begin{theorem}
\label{theorem:continuity-of-homomorphism-c-star}
    Let $A, B$ be unital $C^*$-algebras and $\Phi: A \to B$ be a unital *-homomorphism, then $\Phi(A)$ is closed. 
\end{theorem}
\begin{proof}[Proof, {{\cite[Theorem 11.1]{Zhu}}}. ]
\end{proof}