Added basics of C*-algebras.

src/op/c-star/sa.tex

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+\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}
+
+
+
+