Added quotients.

src/op/c-star/index.tex

@@ -13,3 +13,4 @@
 \input{./state.tex}
 \input{./gns.tex}
 \input{./non-unital.tex}
+\input{./quotients.tex}

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}
 
-

src/op/c-star/quotients.tex

@@ -0,0 +1,54 @@
+\section{Quotients of $C^*$-Algebras}
+\label{section:cstar-quotient}
+
+\begin{lemma}
+\label{lemma:two-sided-ideal-self-adjoint}
+    Let $A$ be a unital $C^*$-algebra and $I \subset A$ be a closed two-sided ideal, then for each $x \in I$, $x^* \in I$ as well. 
+\end{lemma}
+\begin{proof}
+    By \autoref{theorem:cstar-approximate-identity-existence}, there exists an increasing net $\angles{e_\beta}_{\beta \in B} \subset I \cap \ol{B_A(0, 1)}$ such that $e_\beta x \to x$ and $xe_\beta \to x$ for all $x \in I$. As $A$ is a $C^*$-algebra, 
+    \[
+    \norm{e_\beta x - x}_A = \norm{x^*e_\beta - x^*}_A \to 0
+    \]
+
+    Given that $I$ is two-sided, $x^*e_\beta \in I$ for all $\beta \in B$. Since $I$ is closed, the above implies that $x^* \in I$ as well. 
+\end{proof}
+
+
+\begin{lemma}
+\label{lemma:cstar-ideal-quotient-norm}
+    Let $A$ be a unital $C^*$-algebra, $I \subset A$ be a closed two-sided ideal, and $\angles{e_\beta}_{\beta \in B}$ be an increasing approximate identity for $I$, then for each $x \in A$, 
+    \[
+    \norm{x + I}_{A/I} = \inf\bracsn{\norm{x - y}_A|y \in I} = \lim_{\beta \in B}\norm{xe_\beta - x}_A
+    \]
+\end{lemma}
+\begin{proof}[Proof, {{\cite[Lemma 15.6]{Zhu}}}. ]
+    For each $y \in I$, $e_\beta y \to y$. Thus
+    \[
+    \limsup_{\beta \in B}\norm{xe_\beta - x}_A = \limsup_{\beta \in B}\norm{(x - y)(1 - e_\beta)}_A \le \norm{x - y}_A
+    \]
+
+    As the above holds for all $y \in I$, $\limsup_{\beta \in B}\norm{xe_\beta - x}_A \le \norm{x + I}_{A/I}$. 
+
+    On the other hand, since $I$ is a two-sided ideal, $xe_\beta \in I$ for all $\beta \in B$. Thus $\liminf_{\beta \in B}\norm{xe_\beta - x}_A \ge \norm{x + I}_{A/I}$.
+\end{proof}
+
+\begin{theorem}
+\label{theorem:cstar-quotient}
+    Let $A$ be a unital $C^*$-algebra and $I \subset A$ be a closed two-sided ideal, then $A/I$ equipped with the quotient norm is a $C^*$-algebra. 
+\end{theorem}
+\begin{proof}
+    Under the quotient structures, $A/I$ is an involutive Banach algebra. It remains to show that $\norm{x + I}_{A/I}^2 = \norm{x^*x + I}_{A/I}$ for all $x \in A$. 
+
+    By \autoref{lemma:two-sided-ideal-self-adjoint}, $I$ is a $C^*$-algebra. Thus \autoref{theorem:cstar-approximate-identity-existence} implies the existence of an increasing approximate identity $\angles{e_\beta}_{\beta \in B} \subset I$ for $I$. 
+
+    For each $x \in A$ and $y \in I$, 
+    \begin{align*}
+        \norm{x + I}_{A/I}^2 &= \lim_{\beta \in B}\norm{x - e_\beta x}_A^2 = \lim_{\beta \in B}\norm{(x - e_\beta x)(x - e_\beta x)^*}_A \\
+        &= \lim_{\beta \in B}\norm{(1_A - e_\beta)xx^*(1_A - e_\beta)}_A \\ 
+        &= \lim_{\beta \in B}\norm{(1_A - e_\beta)(xx^* + y)(1_A - e_\beta)}_A \le \norm{xx^* + y}_{A}
+    \end{align*}
+
+    As the above holds for all $y \in I$, $\norm{x + I}_{A/I}^2 \le \norm{xx^* + I}_{A}$. Thus $A/I$ equipped with the quotient norm is a $C^*$-algebra. 
+\end{proof}
+