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Bokuan Li
2026-07-07 20:50:19 -04:00
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Let $A$ be a Banach algebra, then $\Omega(A)$ is the \textbf{space of multiplicative linear functionals}, and with respect to the weak-* topology,
\begin{enumerate}
\item If $A$ is unital, then $\Omega(A)$ is a compact Hausdorff space.
\item $\Omega(A) \cup \bracs{0}$ is a compact Hausdorff space, and $\Omega(A)$ is a LCH space.
\item $\Omega(A) \cup \bracs{0}$ is a compact Hausdorff space, and $\Omega(A)$ is an LCH space.
\end{enumerate}
\end{definition}
\begin{proof}