\section{$C_0(X)$}
\label{section:vanishing-infinity-algebra}

\begin{definition}[$C_0(X)$]
\label{definition:vanishing-infinity-algebra}
    Let $X$ be a LCH space, then $C_0(X; \complex)$ equipped with pointwise operations and the uniform norm is a $C^*$-algebra. 
\end{definition}

\begin{theorem}
\label{theorem:vanishing-infinity-multiplicative-functional}
    Let $X$ be a LCH space, then the mapping
    \[
    E: X \to \Omega(C_0(X)) \quad E(x)(f) = f(x)
    \]

    is a homeomorphism. Under the identification $X = \Omega(C_0(X))$, the Gelfand transform is the identity. 
\end{theorem}
\begin{proof}[Proof, {{\cite[Theorem 7.4]{Zhu}}}. ]
    Let $X^* = X \sqcup \bracs{\infty}$ be the \hyperref[one-point compactification]{definition:alexandroff-compactification} of $X$. For each $\phi \in \Omega(C_0(X))$, let
    \[
    \phi^*: BC(X^*) \to \complex \quad f \mapsto \phi(f - f(\infty)) + f(\infty)
    \]

    then for each $f, g \in BC(X^*)$, 
    \begin{align*}
        fg &= (f - f(\infty))(g - g(\infty)) + f(\infty)(g - g(\infty)) + g(\infty)(f - f(\infty)) + f(\infty)g(\infty) \\
        \phi^*(fg) &= \phi(f - f(\infty))\phi(g - g(\infty)) + f(\infty)\phi(g - g(\infty)) \\
        &+ g(\infty)(f - f(\infty)) + f(\infty)g(\infty) \\
        &= \braksn{\phi(f - f(\infty)) + f(\infty)}\braksn{\phi(g - g(\infty)) + g(\infty)} \\
        &= \phi^*(f)\phi^*(g)
    \end{align*}

    so $\phi^* \in \Omega(BC(X^*))$. By \autoref{theorem:multiplicative-functional-bc}, there exists $x \in X^*$ such that $\phi^*(f) = f(x)$ for all $f \in BC(X^*)$. Since $\phi \ne 0$, $x \in X$, and $\phi = E(x)$. 
\end{proof}

