Added a section on non-unital C^*-algebras.

src/op/c-star/order.tex

@@ -82,6 +82,19 @@
     The condition in the sign decomposition that $x^+x^- = x^-x^+ = 0$ is essential. Otherwise I may use silly decompositions like $0 = 1 - 1$. 
 \end{remark}
 
+\begin{lemma}
+\label{lemma:cstar-inversion-order-reversing}
+    Let $A$ be a unital $C^*$-algebra, $x, y \in G(A)$ be positive elements with $x \le y$, then $x^{-1} \ge y^{-1}$.
+\end{lemma}
+\begin{proof}
+    Since $y - x \ge 0$ and $x$ is invertible, $y^{-1/2}(y - x)y^{-1/2} \ge 0$ as well. As such, 
+    \[
+    y^{-1/2}xy^{-1/2} \le y^{-1/2}yy^{-1/2} = 1
+    \]
+
+    Thus $\sigma_A(y^{-1/2}xy^{-1/2}) \subset \ol{B_\complex(0, 1)}$, and $\sigma_A(y^{1/2}x^{-1}y^{1/2}) \subset \complex \setminus B_\complex(0, 1)$. Hence $y^{1/2}x^{-1}y^{1/2} \ge 1$, and $x^{-1} \ge y^{-1}$. 
+\end{proof}
+