36 lines
1.6 KiB
TeX
36 lines
1.6 KiB
TeX
\section{$C_0(X)$}
|
|
\label{section:vanishing-infinity-algebra}
|
|
|
|
\begin{definition}[$C_0(X)$]
|
|
\label{definition:vanishing-infinity-algebra}
|
|
Let $X$ be an 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 an 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}
|
|
|