Added quotients.

src/op/c-star/non-unital.tex

@@ -96,7 +96,7 @@
     
     then $\angles{e_F}_{F \in \cf} \subset I \cap \ol{B_A(0, 1)}$ is an increasing net of positive elements such that $xe_F \to x$ for all $x \in I$. 
 \end{theorem}
-\begin{proof}
+\begin{proof}[Proof, {{\cite[Theorem 15.2]{Zhu}}}. ]
     Since $p_F$ is positive, $1/|F| + p_F$ is invertible by \autoref{proposition:positive-spectrum}, and the \hyperref[continuous functional calculus]{definition:continuous-functional-calculus} implies that $0 \le e_F \le 1_A$. As $I \subset A$ is a left ideal, $e_F \in I \cap \ol{B_A(0, 1)}$. Now,
     \begin{align*}
         &\sum_{x \in F}[x(e_F - 1_A)]^*[x(e_F - 1_A)] = \sum_{x \in F}(e_F - 1_A)x^*x(e_F - 1_A) \\
@@ -149,4 +149,3 @@
     Identify $A$ as a self-adjoint two-sided ideal of its unitisation $\td A$, which is a $C^*$-algebra by \autoref{proposition:c-star-unitisation}. Applying \autoref{theorem:cstar-approximate-identity-existence} to $A$ and $\bracs{x^*|x \in A}$ as left and right ideals, respectively, yields that the net constructed by the theorem is an increasing approximate identity for $A$. 
 \end{proof}
 
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