Added basics of C*-algebras.

src/fa/tvs/complexify.tex

@@ -58,8 +58,8 @@
 \label{definition:complex-conjugation}
     Let $E$ be a vector space over $\complex$ and $*: E \to E$ be a $\real$-linear map, then $*$ is a \textbf{complex conjugation} if:
     \begin{enumerate}
-        \item For each $\lambda \in \complex$, $(\lambda x)^* = \ol \lambda x^*$. 
-        \item For each $x \in E$, $x^{**} = x$. 
+        \item[(C1)] For each $\lambda \in \complex$, $(\lambda x)^* = \ol \lambda x^*$. 
+        \item[(C2)] For each $x \in E$, $x^{**} = x$. 
     \end{enumerate}
 
     In which case, $\text{Re}(E) = \bracs{x \in E| x^* = x}$ is the \textbf{real part} of $E$. 
@@ -74,6 +74,7 @@
         \[
         \text{Re}(x) = \frac{x + x^*}{2} \quad \text{Im}(x) = \frac{x - x^*}{2i}
         \]
+        \item For each $x \in E$, $x^* = \text{Re}(x) - i\text{Im}(x)$. 
     \end{enumerate}
 \end{proposition}
 \begin{proof}

src/op/c-star/index.tex

@@ -0,0 +1,6 @@
+\chapter{$C^*$-Algebras}
+\label{chap:c-star-algebras}
+
+\input{./involution.tex}
+\input{./sa.tex}
+\input{./order.tex}

src/op/c-star/involution.tex

@@ -0,0 +1,57 @@
+\section{Involutions}
+\label{section:involutions}
+
+\begin{definition}[Involution]
+\label{definition:involution}
+    Let $A$ be an associative algebra over $\complex$, and $*: A \to A$ be a $\real$-linear map, then $*$ is an \textbf{involution} if:
+    \begin{enumerate}
+        \item[(C1)] For each $\lambda \in \complex$, $(\lambda x)^* = \ol \lambda x^*$. 
+        \item[(C2)] For each $x \in A$, $x^{**} = x$. 
+        \item[(I)] For every $x, y \in A$, $(xy)^* = y^*x^*$.  
+    \end{enumerate}
+
+    The space $A$ equipped with an involution is an \textbf{involutive algebra} over $\complex$. 
+\end{definition}
+
+\begin{definition}[$C^*$-Algebra]
+\label{definition:c-star-algebra}
+    Let $A$ be an involutive Banach algebra over $\complex$, then $A$ is a \textbf{$C^*$-algebra} if for every $x \in A$, $\normn{x^*x}_A = \norm{x}_A^2$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:c-star-algebra-gymnastics}
+    Let $A$ be a $C^*$ algebra, then:
+    \begin{enumerate}
+        \item For each $x \in A$, $\norm{x}_A = \normn{x^*}_A\norm{x}_A$. 
+    \end{enumerate}
+
+    If $A$ is unital, then
+    \begin{enumerate}[start=1]
+        \item For each $\lambda \in \complex$, $\lambda^* = \ol \lambda$. 
+        \item For any $x \in A$, $x \in G(A)$ if and only if $x^* \in G(A)$.  
+        \item For every $x \in A$, $\sigma_A(x^*) = \bracsn{\ol \lambda| \lambda \in \sigma_A(x)}$.
+        \item For each $x \in A$, $[x]_{sp} = [x^*]_{sp}$. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1): For each $x \in A$, $\norm{x}_A^2 = \normn{x^*x}_A \le \norm{x}_A \normn{x^*}_A$. 
+
+    (2): For every $x \in A$, $1^*x^* = (x1)^* = x^* = (1x)^* = x^*1^*$, so $1^* = 1$ by uniqueness of the inverse. 
+
+    (3): For any $x \in A$, $(x^{-1})^*x^* = (x^{-1}x)^* = 1 = (xx^{-1})^* = x^*(x^{-1})^*$.
+\end{proof}
+
+\begin{definition}[*-Homomorphism]
+\label{definition:star-homomorphism}
+    Let $A, B$ be $C^*$-algebras and $\phi: A \to B$, then $\phi$ is a \textbf{*-homomorphism} if:
+    \begin{enumerate}
+        \item $\phi$ is a homomorphism of Banach algebras.
+        \item For every $x \in A$, $\phi(x^*) = \phi(x)^*$. 
+    \end{enumerate}
+
+    If in addition, $\phi(1) = 1$, then $\phi$ is a \textbf{unital *-homomorphism}. 
+\end{definition}
+
+
+
+

src/op/c-star/order.tex

@@ -0,0 +1,10 @@
+\section{Order Structures of $C^*$-Algebras}
+\label{section:order-c-star-algebra}
+
+\begin{definition}[Positive]
+\label{definition:positive-c-star-algebra}
+    Let $A$ be a $C^*$-algebra and $x \in A$, then $x$ is \textbf{positive} if there exists $y \in A$ such that $x = y^*y$. 
+\end{definition}
+
+
+

src/op/c-star/sa.tex

@@ -0,0 +1,36 @@
+\section{Self-Adjoint Elements}
+\label{section:c-star-self-adjoint}
+
+\begin{definition}[Self-Adjoint]
+\label{definition:self-adjoint}
+    Let $A$ be an involutive algebra over $\complex$ and $x \in A$, then $x$ is \textbf{self-adjoint} if $x = x^*$. The space $A_{sa} = \bracs{x \in A| x = x^*}$ is the \textbf{self-adjoint part} of $A$, and:
+    \begin{enumerate}
+        \item $A_{sa}$ is a $\real$ subspace of $A$. 
+        \item $A = \complex(A_{sa})$ as a vector space.
+        \item For each $x \in A$, let
+        \[
+        \text{Re}(x) = \frac{x + x^*}{2} \quad \text{Im}(x) = \frac{x - x^*}{2i}
+        \]
+        
+        then $\text{Re}(x), \text{Im}(x) \in A_{sa}^2$ and $x = \text{Re}(x) + i\text{Im}(x)$.
+        \item For each $x \in A$, $x^* = \text{Re}(x) - i\text{Im}(x)$. 
+    \end{enumerate}
+\end{definition}
+\begin{proof}
+    By \autoref{proposition:complex-conjugation-properties}. 
+\end{proof}
+
+\begin{definition}[Normal]
+\label{definition:c-star-normal}
+    Let $A$ be an involutive algebra over $\complex$ and $x \in A$, then the following are equivalent:
+    \begin{enumerate}
+        \item $\text{Re}(x)\text{Im}(x) = \text{Im}(x)\text{Re}(x)$. 
+        \item $x^*x = xx^*$. 
+    \end{enumerate}
+    
+    If the above holds, then $x$ is \textbf{normal}. 
+\end{definition}
+
+
+
+

src/op/index.tex

@@ -2,5 +2,6 @@
 \label{part:operator-algebras}
 
 \input{./banach/index.tex}
+\input{./c-star/index.tex}
 \input{./example/index.tex}
 \input{./notation.tex}