Added a handful of examples.

src/op/banach/igroup.tex

@@ -26,10 +26,15 @@
     \sigma_A(x) \subset \complex \setminus e^{i\theta}[0, \infty)
     \]
 
-    then $x \in G_0(A)$. 
+    \begin{enumerate}
+        \item There exists $y \in A$ such that $x = \exp(y)$. 
+        \item $x \in G_0(A)$. 
+    \end{enumerate}
 \end{proposition}
 \begin{proof}
-    Since there exists a branch of the logarithm $\ell: \complex \setminus e^{i\theta}[0, \infty) \to \complex$, there exists $y \in A$ such that $x = \exp(y)$. In which case, $x \in G_0(A)$ by \autoref{proposition:functional-calculus-exp}. 
+    (1): Since there exists a branch of the logarithm $\ell: \complex \setminus e^{i\theta}[0, \infty) \to \complex$, there exists $y \in A$ such that $x = \exp(y)$. 
+    
+    (2): \autoref{proposition:functional-calculus-exp}. 
 \end{proof}
 
 

src/op/example/bounded.tex

@@ -0,0 +1,11 @@
+\section{$B(H)$}
+\label{section:hilbert-endomorphism}
+
+\begin{definition}[$B(H)$]
+\label{definition:hilbert-endomorphism}
+    Let $H$ be a Hilbert space, then $B(H) = L(H; H)$ is the algebra of all bounded linear operators on $H$. 
+\end{definition}
+
+
+% 1. Every non-trivial ideal of B(H) contains the finite-rank operators.
+% 2. If H is separable, then the only non-trivial closed idela of B(H) are the compact operators. 

src/op/example/disk.tex

@@ -0,0 +1,22 @@
+\section{The Disk Algebra}
+\label{section:disk-algebra}
+
+\begin{definition}[Disk Algebra]
+\label{definition:disk-algebra}
+    Let $D = B_\complex(0, 1)$, then the \textbf{disk algebra} $A(D) = H(D; \complex) \cap C(\ol D; \complex)$ is the space of holomorphic functions on $D$ that extend to $\ol D$, equipped with the uniform norm. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:disk-algebra-spectrum}
+    Let $f \in A(D)$, then $\sigma(f) = f(\ol D)$. 
+\end{proposition}
+
+\begin{proposition}
+\label{proposition:disk-algebra-multiplicative-functional}
+    Let $\phi \in A(D)^*$ be a multiplicative linear functional, then there exists $z_0 \in \ol D$ such that for every $f \in A(D)$, $\phi(f) = f(z_0)$. 
+\end{proposition}
+\begin{proof}
+    By \autoref{proposition:multiplicative-unit}, $\norm{\phi}_{A(D)^*} = 1$. Let $p$ be the identity polynomial, and $z_0 = \phi(p)$, then for every $q \in \complex[x]$, $\phi(q) = q(x_0)$. By density of polynomials in $A(D)$, $\phi(f) = f(z_0)$ for all $f \in A(D)$. 
+\end{proof}
+
+

src/op/example/hardy.tex

@@ -0,0 +1,35 @@
+\section{The Hardy Space}
+\label{section:hardy-space}
+
+\begin{definition}[Hardy Space]
+\label{definition:hardy-space}
+    Let $D = B_\complex(0, 1)$, then the \textbf{Hardy space} $H^\infty(D)$ is the space of all bounded holomorphic functions on $D$, equipped with the uniform norm. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:hardy-spectrum}
+    Let $f \in H^\infty(D)$, then $\sigma(f) = \ol{f(D)}$. 
+\end{proposition}
+
+\begin{proposition}
+\label{proposition:hardy-non-trivial-functional}
+    Let $\fU \subset 2^{B_\complex(0, 1)}$ be an ultrafilter and
+    \[
+    \phi_{\fU}: H^\infty(D) \to \complex \quad f \mapsto \lim_{x, \fU} f(x)
+    \]
+
+    then:
+    \begin{enumerate}
+        \item If $\fU \to x_0 \in \ol{D}$, then $f \in A(D)$, $\phi_{\fU}(f) = f(z_0)$. 
+        \item $\phi_{\fU}$ is a multiplicative linear functional on $H^\infty(D)$. 
+    \end{enumerate}
+    
+\end{proposition}
+\begin{proof}
+    Let $f \in H^\infty(D)$, then by \autoref{proposition:imagefilterbase}, $f(\fU)$ is an ultrafilter base. Since $f$ is bounded, $f(\fU)$ converges to exactly one element of $\complex$. Hence the limit is well-defined. 
+
+    (2): By \autoref{proposition:operator-space-completeness}. 
+\end{proof}
+
+
+

src/op/example/index.tex

@@ -0,0 +1,8 @@
+\chapter{Examples of Operator Algebras}
+\label{chap:op-examples}
+
+\input{./matrix.tex}
+\input{./bounded.tex}
+\input{./hardy.tex}
+\input{./disk.tex}
+

src/op/example/matrix.tex

@@ -0,0 +1,44 @@
+\section{The Matrix Algebra}
+\label{section:matrix-algebra}
+
+\begin{definition}[Matrix Algebra]
+\label{definition:matrix-algebra}
+    Let $n \in \natp$ and $M_n(\complex)$ be the set of all $n \times n$ matrices with entries in $\complex$, then $M_n(\complex)$ equipped with the operator norm is the \textbf{matrix algebra} over $\complex$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:matrix-algebra-spectrum}
+    Let $n \in \natp$ and $x \in M_n(\complex)$, then $\sigma(x)$ is the set of eigenvalues of $x$. 
+\end{proposition}
+\begin{proof}
+    By the rank-nullity theorem, $\lambda - x$ is invertible if and only if $\lambda$ is not an eigenvalue of $x$. 
+\end{proof}
+
+\begin{proposition}
+\label{proposition:matrix-algebra-index}
+    Let $n \in \natp$, then
+    \begin{enumerate}
+        \item For each $x \in G(M_n(\complex))$, there exists $y \in M_n(\complex)$ such that $x = \exp(y)$. 
+        \item $G(M_n(\complex)) = G_0(M_n(\complex))$. 
+        \item $I(M_n(\complex))$ is trivial. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1), (2): By \autoref{proposition:matrix-algebra-spectrum}, $\sigma(x)$ is finite. By \autoref{proposition:log-identity-component}, there exists $y \in M_n(\complex)$ such that $x = \exp(y)$. In which case, $x \in G_0(M_n(\complex))$. 
+\end{proof}
+
+\begin{proposition}
+\label{proposition:matrix-algebra-ideals}
+    Let $n \in \natp$, then
+    \begin{enumerate}
+        \item $M_n(\complex)$ admits no nontrivial two-sided ideals. 
+        \item $M_n(\complex)$ admits no multiplicative functionals. 
+    \end{enumerate}
+    
+\end{proposition}
+\begin{proof}
+    Let $x = (x_{ij}) \in M_n(\complex) \setminus \bracs{0}$, then there exists $1 \le i, j \le n$ such that $x_{ij} \ne 0$. In which case, for any $1 \le k, l \le n$ and $\lambda \in \complex$, there exists $y_k, z_l \in M_n(\complex)$ such that $y_kxz_l$ is the matrix with $\lambda$ on its $(k, l)$ entry and $0$ everywhere else. Therefore every non-trivial two-sided ideal of $M_n(\complex)$ is $M_n(\complex)$. 
+\end{proof}
+
+
+

src/op/index.tex

@@ -2,4 +2,5 @@
 \label{part:operator-algebras}
 
 \input{./banach/index.tex}
+\input{./example/index.tex}
 \input{./notation.tex}

src/op/notation.tex

@@ -12,5 +12,9 @@
     $R_x(\lambda)$ & The resolvent of $x$. & \autoref{definition:resolvent} \\ 
     $[x]_{sp}$ & The spectral radius of $x$. & \autoref{definition:spectral-radius} \\
     $\Omega(A)$ & Space of multiplicative functionals on $A$. & \autoref{definition:multiplicative-functional} \\
-    $\cm(A)$ & Maximal ideal space of $A$. & \autoref{definition:maximal-ideal}
+    $\cm(A)$ & Maximal ideal space of $A$. & \autoref{definition:maximal-ideal} \\
+    $M_n(\complex)$ & Algebra of $n \times n$ matrices over $\complex$.  & \autoref{definition:matrix-algebra} \\
+    $B(H)$ & Algebra of bounded operators on a Hilbert space. & \autoref{definition:hilbert-endomorphism} \\
+    $A(D)$ & The disk algebra. & \autoref{definition:disk-algebra} \\
+    $H^\infty(D)$ & The Hardy space.  & \autoref{definition:hardy-space} \\
 \end{tabular}