Added basic facts on maximal ideals.

src/op/banach/definitions.tex

@@ -12,8 +12,11 @@
 
 \begin{definition}[Unital Banach Algebra]
 \label{definition:unital-banach-algebra}
-    Let $A$ be a Banach algebra, then $A$ is \textbf{unital} if there exists $1 \in A$ such that for any $x \in A$, $x1 = 1x = x$. In which case, $1$ is the unique \textbf{multiplicative identity} of $A$. 
+    Let $(A, \norm{\cdot}_A)$ be a Banach algebra, then $A$ is \textbf{unital} if there exists $1 \in A$ such that for any $x \in A$, $x1 = 1x = x$. In which case, there exists an equivalent norm $\norm{\cdot}_{1}: A \to [0, \inftu)$ such that $\norm{1}_1 = 1$, and $A$ is always assumed to be equipped with this norm.  
 \end{definition}
+\begin{proof}[Proof, {{\cite[Proposition I.1.3]{Takesaki1}}}. ]
+    For each $x \in A$, let $L_x \in L(A; A)$ be defined by $y \mapsto xy$, and let $\norm{x}_1 = \norm{L_x}_{L(A; A)}$, then $\norm{x}_1 \le \norm{x}_A$ and $\norm{1}_1 = 1$. On the other hand, $\frac{\norm{x}_A}{\norm{1}_A} \le \norm{x}_1$, so $\norm{\cdot}_1$ is equivalent to $\norm{\cdot}_A$.
+\end{proof}
 
 \begin{definition}[Homomorphism]
 \label{definition:banach-algebra-homomorphism}

src/op/banach/ideal.tex

@@ -0,0 +1,59 @@
+\section{Ideals}
+\label{section:banach-algebra-ideals}
+
+\begin{definition}[Ideal]
+\label{definition:banach-algebra-ideal}
+    Let $A$ be a Banach algebra and $I \subset A$, then $I$ is a \textbf{left ideal} if:
+    \begin{enumerate}
+        \item $I$ is a subspace of $A$. 
+        \item For each $x \in A$, $aI \subset I$. 
+    \end{enumerate}
+
+    and $I$ is a \textbf{two-sided ideal} if in addition to the above, 
+    \begin{enumerate}[start=2]
+        \item For each $x \in A$, $Ia \subset I$. 
+    \end{enumerate}
+
+    \textit{In this document, the "sidedness" of ideals are omitted. Within each block, the statement should hold as long as the interpretation is consistent. }
+\end{definition}
+
+\begin{definition}[Proper Ideal]
+\label{definition:banach-algebra-proper-ideal}
+    Let $A$ be a Banach algebra and $I \subset A$ be an ideal, then $I$ is \textbf{proper} if $I \subsetneq A$.  
+\end{definition}
+
+\begin{definition}[Maximal Ideal]
+\label{definition:maximal-ideal}
+    Let $A$ be a Banach algebra and $I \subset A$ be an ideal, then $I$ is \textbf{maximal} if:
+    \begin{enumerate}
+        \item $I$ is proper.
+        \item For any proper ideal $J \subsetneq A$ with $J \supset I$, $I = J$. 
+    \end{enumerate}
+
+    The set $\cm(A)$ of maximal two-sided ideals of $A$ is the \textbf{maximal ideal space} of $A$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:ideal-gymnastics}
+    Let $A$ be a unital Banach algebra and $I \subset A$ be a proper ideal, then:
+    \begin{enumerate}
+        \item $I$ contains no invertible elements. 
+        \item $\ol I$ is also a proper ideal.
+        \item $I$ is contained in a maximal ideal.
+        \item If $I$ is maximal, then $I$ is closed. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}[Proof, {{\cite{FollandHarmonic}}}. ]
+    (1): If $I$ contains an invertible element, then $1 \in I$ and thus $A = I$. 
+
+    (2): By \autoref{proposition:banach-algebra-inverse}, $G(A)$ is open. Since $G(A) \cap I = \emptyset$, $G(A) \cap \ol I = \emptyset$ as well. 
+
+    (3): By Zorn's lemma. 
+
+    (4): By (2). 
+\end{proof}
+
+
+
+
+

src/op/banach/index.tex

@@ -4,5 +4,7 @@
 \input{./definitions.tex}
 \input{./invertible.tex}
 \input{./igroup.tex}
+\input{./ideal.tex}
 \input{./spectrum.tex}
 \input{./fc.tex}
+\input{./multiplicative.tex}

src/op/banach/multiplicative.tex

@@ -0,0 +1,50 @@
+\section{Multiplicative Functionals}
+\label{section:multiplicative-functional}
+
+\begin{definition}[Multiplicative Functional]
+\label{definition:multiplicative-functional}
+    Let $A$ be a unital Banach algebra and $\phi \in A^*$, then $\phi$ is \textbf{multiplicative} if $\phi \ne 0$ and for each $x, y \in A$, $\phi(xy) = \phi(x)\phi(y)$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:multiplicative-unit}
+    Let $A$ be a unital Banach algebra and $\phi \in A^*$ be a multiplicative functional, then
+    \begin{enumerate}
+        \item For each $x \in A$, $|\phi(x)| \le [x]_{sp} \le \norm{x}_A$. 
+        \item $\norm{\phi}_{A^*} = 1$. 
+        \item $\phi(G(A)) \subset \complex \setminus \bracs{0}$. 
+    \end{enumerate}
+    
+\end{proposition}
+\begin{proof}
+    (3): For each $x \in G(A)$, $1 = \phi(xx^{-1}) = \phi(x)\phi(x^{-1}) \ne 0$. 
+    
+    (1): By (3), for every $\lambda \in \complex$ with $|\lambda| > [x]_{sp}$, $\lambda x \in G(A)$ and $\lambda - \phi(x) \ne 0$. Therefore $\phi(x) \in \ol{B(0, [x]_{sp})}$. 
+
+    (2): For each $\lambda \in \complex$, $\phi(\lambda 1) = \lambda$, so $\norm{\phi}_{A^*} \le 1$. 
+\end{proof}
+
+\begin{definition}[Space of Multiplicative Linear Functionals]
+\label{definition:multiplicative-linear-functional-space}
+    Let $A$ be a unital Banach algebra, then $\Omega(A)$ is the \textbf{space of multiplicative linear functionals}, which is a compact Hausdorff space under the weak-* topology. 
+\end{definition}
+\begin{proof}
+    By the \hyperref[Banach-Alaoglu Theorem]{theorem:alaoglu}.
+\end{proof}
+
+\begin{proposition}
+\label{proposition:commutative-maximal-ideal-space}
+    Let $A$ be a commutative unital Banach algebra, then the mapping
+    \[
+    \Omega(A) \to \cm(A) \quad \phi \mapsto \ker(\phi)
+    \]
+
+    is a bijection. 
+\end{proposition}
+\begin{proof}
+    For each $\phi \in \cm(A)$, $\ker(\phi)$ is an ideal of codimension $1$, and must be maximal. 
+
+    On the other hand, for each $I \in \Omega(A)$, the quotient $A/I$ is a field, so by the \hyperref[Gelfand-Mazur Theorem]{theorem:gelfand-mazur}, it is isomorphic to $\complex$. Thus the canonical projection $A \to A/I \iso \complex$ induces a multiplicative functional on $A$. 
+\end{proof}
+
+

src/op/notation.tex

@@ -10,5 +10,7 @@
     $I(A)$ & The index group of $A$. & \autoref{definition:index-group} \\
     $\sigma_A(x) = \sigma(x)$ & The spectrum of $x$ in $A$. & \autoref{definition:spectrum} \\
     $R_x(\lambda)$ & The resolvent of $x$. & \autoref{definition:resolvent} \\ 
-    $[x]_{sp}$ & The spectral radius of $x$. & \autoref{definition:spectral-radius}
+    $[x]_{sp}$ & 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}
 \end{tabular}