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}