Adjusted prose in example facts.

src/op/example/bc.tex

@@ -8,8 +8,6 @@
     Equipped with pointwise operations and the uniform norm, $BC(X; \complex)$ forms a commutative Banach algebra. 
 \end{definition}
 
-
-
 \begin{theorem}
 \label{theorem:multiplicative-functional-bc}
     Let $X$ be a completely regular space and $\beta X$ be its \hyperref[Stone-Čech compactification]{definition:stone-cech}. For each $f \in BC(X; \complex)$, let $\beta f \in BC(\beta X; \complex)$ be its unique extension to $\beta X$, then the mapping
@@ -17,7 +15,7 @@
     E: \beta X \to \Omega(BC(X; \complex)) \quad E(x)(f) = (\beta f)(x)
     \]
 
-    is a homeomorphism. 
+    is a homeomorphism. Under the identification $\beta X = \Omega(BC(X; \complex))$, $\Gamma_{BC(X; \complex)} = \beta$. 
 \end{theorem}
 \begin{proof}
     Let $\phi \in BC(X; \complex)^* \setminus \ol{E(X)}$, then there exists $\seqf{f_k} \subset BC(X; \complex)$ and $\eps > 0$ such that for every $x \in X$, 

src/op/example/convolution.tex

@@ -53,7 +53,10 @@
     (F(z))(f) = \sum_{n \in \integer}f(n)z^n
     \]
 
-    is a homeomorphism. 
+    is a homeomorphism. Under the identification $\partial B_\complex(0, 1) = \Omega(\ell^1(\integer))$, the Gelfand transform has the explicit form
+    \[
+    \Gamma(f)(z) = \sum_{n \in \integer} f(n)z^n
+    \]
 \end{theorem}
 \begin{proof}{Proof, {{\cite[Theorem 6.3]{Zhu}}}. }
     Let $z \in \mathbf{S}^1$ and $f, g \in \ell^1(\integer)$, then by \hyperref[Fubini's Theorem]{theorem:fubini-tonelli},
@@ -70,6 +73,15 @@
     Finally, by the \hyperref[Dominated Convergence Theorem]{theorem:dct}, $F: \textbf{S}^1 \to \Omega(\ell^1(\integer))$ is continuous. Since $\textbf{S}^1$ is compact, $\Omega(\ell^1(\integer))$ is Hausdorff, and $F$ is a bijection, $F$ is a homeomorphism by \autoref{proposition:compact-hausdorff-homeomorphism}. 
 \end{proof}
 
-
+\begin{proposition}
+\label{proposition:convolution-integer-spectrum}
+    Let $\ell^1(\integer)$ be the convolution algebra on $\integer$ and $f \in \ell^1(\integer)$, then
+    \[
+    \sigma(f) = \bracs{\sum_{n \in \integer}f(n)z^n \bigg | z \in \partial B_\complex(0, 1)}
+    \]
+\end{proposition}
+\begin{proof}
+    By \autoref{theorem:convolution-integer-gelfand} and (4) of \autoref{proposition:gelfand-transform-gymnastics}.
+\end{proof}
 
 

src/op/example/index.tex

@@ -6,4 +6,4 @@
 \input{./hardy.tex}
 \input{./disk.tex}
 \input{./convolution.tex}
-
+\input{./bc.tex}