Added Gelfand Naimark.

src/op/c-star/gelfand.tex

@@ -0,0 +1,77 @@
+\section{The Gelfand-Naimark Theorem}
+\label{section:gelfand-naimark}
+
+\begin{theorem}[Gelfand-Naimark]
+\label{theorem:gelfand-naimark}
+    Let $A$ be a commutative unital $C^*$-algebra, then the Gelfand transform
+    \[
+    \Gamma_A: A \to C(\Omega(A); \complex) \quad \Gamma_A(x)(\phi) = \phi(x)
+    \]
+
+    is a unital $C^*$-isomorphism. 
+\end{theorem}
+\begin{proof}[Proof, {{\cite[Theorem II.9.4]{Zhu}}}. ]
+    By construction $\Gamma_A$ is a unital algebra homomorphism. 
+
+    To see that $\Gamma_A$ preserves involutions, let $y \in A$ be self-adjoint. By \autoref{proposition:gelfand-transform-gymnastics} and \autoref{proposition:self-adjoint-spectrum}, $\Gamma_A(y)(\Omega(A)) = \sigma_A(y) \subset \real$, so $\Gamma_A(y) \in C(\Omega(A); \real)$. For any $x \in A$, write $x = \text{Re}(x) + i\text{Im}(x)$, where $\text{Re}(x)$ and $\text{Im}(x)$ are both self-adjoint, then
+    \begin{align*}
+        \Gamma_A(x^*) &= \Gamma_A(\text{Re}(x) - i\text{Im}(x)) \\
+        &= \Gamma_A(\text{Re}(x)) - i\Gamma_A(\text{Im}(x)) \\
+        &= \overline{\Gamma_A(\text{Re}(x)) + i\Gamma_A(\text{Im}(x))} = \overline{\Gamma_A(x)}
+    \end{align*}
+
+    so $\Gamma_A(x^*) = \Gamma_A(x)^*$. 
+
+    Now, for each $x \in A$, \autoref{corollary:c-star-unique-norm} and \autoref{proposition:gelfand-transform-gymnastics} imply that
+    \begin{align*}
+        \norm{x}_A^2 &= \sup\bracs{|\lambda|\ | \lambda \in \sigma_A(x^*x)} \\
+        &= \sup\bracs{|\Gamma_A(x^*x)(\phi)|\ | \phi \in \Omega(A)} \\
+        &= \sup\bracs{|\Gamma_A(x)(\phi)|^2\ | \phi \in \Omega(A)} \\
+        \norm{x}_A &= \norm{\Gamma_A(x)}_u
+    \end{align*}
+
+    Thus $\Gamma_A$ is an isometry, and $\Gamma_A(A)$ is a closed subalgebra of $C(\Omega(A))$. 
+
+    Since $\Gamma_A(1_A) = 1$, $\Gamma_A(A)$ contains constants. As $\Gamma_A(A)$ separates points and is closed under complex conjugation, $\Gamma_A(A) = C(\Omega(A))$ by the \hyperref[Stone-Weierstrass Theorem]{theorem:complex-stone-weierstrass}.
+\end{proof}
+
+\begin{corollary}
+\label{corollary:gelfand-naimark-converse}
+    Let $A$ be a unital $C^*$-algebra, then the following are equivalent:
+    \begin{enumerate}
+        \item $A$ is commutative. 
+        \item $\Gamma_A$ is a *-isomorphism. 
+        \item $\Gamma_A$ is injective.
+    \end{enumerate}
+\end{corollary}
+
+\begin{corollary}
+\label{corollary:spectrum-characterisation-iff}
+    Let $A$ be a commutative unital $C^*$-algebra and $x \in A$ be normal, then:
+    \begin{enumerate}
+        \item $x$ is self-adjoint if and only if $\sigma_A(x) \subset \real$. 
+        \item $x$ is unitary if and only if $\sigma_A(x) \subset \partial B_\complex(0, 1)$.
+        \item $x$ is positive if and only if $\sigma_A(x) \subset [0, \infty)$. 
+        \item $x$ is a projection if and only if $\sigma_A(x) \subset \bracs{0,1}$. 
+    \end{enumerate}
+\end{corollary}
+% NEEDS WORK
+
+\begin{corollary}
+\label{corollary:stonean-commutative-algebra}
+    Let $A$ be a unital $C^*$-algebra, then $A_{sa}$ is order complete if and only if $\Omega(A)$ is extremely disconnected. 
+\end{corollary}
+\begin{proof}
+    By \autoref{theorem:gelfand-naimark}, $A$ and $C(\Omega(A); \complex)$ are isomorphic as $C^*$-algebras. In particular, $A_{sa}$ and $C(\Omega(A); \real)$ are isomorphic as ordered vector spaces, so $A_{sa}$ is order complete if and only if $C(\Omega(A); \real)$ is order complete. Thus the \hyperref[Stone-Nakano Theorem]{theorem:stone-nakano-extremely-disconnected} implies that $A_{sa}$ is order complete if and only if $\Omega(A)$ is extremely disconnected.
+\end{proof}
+
+\begin{corollary}
+\label{corollary:linfinity-extremely-disconnected}
+    Let $(X, \cm, \mu)$ be a localisable measure space, then $\Omega(L^\infty(X))$ is extremely disconnected. 
+\end{corollary}
+\begin{proof}
+    By \autoref{corollary:l-infty-dedekind-complete}, $L^\infty(X; \real)$ is order complete. By \autoref{corollary:stonean-commutative-algebra}, $\Omega(L^\infty(X))$ is extremely disconnected. 
+\end{proof}
+
+
+

src/op/c-star/homomorphism.tex

@@ -0,0 +1,39 @@
+\section{*-Homomorphisms}
+\label{section:c-star-homomorphism}
+
+\begin{definition}[*-Homomorphism]
+\label{definition:c-star-homomorphism}
+    Let $A, B$ be involutive algebras over $\complex$ and $\phi: A \to B$, then $\phi$ is a \textbf{*-homomorphism} if:
+    \begin{enumerate}[label=(SH\arabic*)]
+        \item For each $x, y \in A$ and $\lambda \in \complex$, $\phi(\lambda x + y) = \lambda \phi(x) + \phi(y)$. 
+        \item For each $x, y \in A$, $\phi(xy) = \phi(x)\phi(y)$. 
+        \item For each $x \in A$, $\phi(x^*) = \phi(x)^*$.
+    \end{enumerate}
+
+    If $A$ and $B$ are unital, then $\phi$ is \textbf{unital} if:
+    \begin{enumerate}
+        \item[(U)] $\phi(1_A) = \one_B$. 
+    \end{enumerate}
+\end{definition}
+
+
+\begin{proposition}
+\label{proposition:star-homomorphism-contractive}
+    Let $A, B$ be unital $C^*$-algebras and $\phi: A \to B$ be a unital *-homomorphism, then for each $x \in A$, 
+    \begin{enumerate}
+        \item $\sigma_B(\phi(x)) \subset \sigma_A(x)$. 
+        \item $\norm{\phi(x)}_B \le \norm{x}_A$. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1): Since $\phi$ is unital, $\phi(G(A)) \subset G(B)$, so $\sigma_B(\phi(x)) \subset \sigma_A(x)$. 
+    
+    (2): By (1) and \autoref{corollary:c-star-unique-norm}, 
+    \begin{align*}
+    \norm{\phi(x)}_B^2 &= \sup\bracsn{|\lambda|\ | \lambda \in \sigma_B(\phi(x^*x))} \\
+    &\ge \sup\bracsn{|\lambda|\ | \lambda \in \sigma_A(x^*x)} = \norm{x}_A^2
+    \end{align*}
+    
+\end{proof}
+
+

src/op/c-star/index.tex

@@ -5,3 +5,5 @@
 \input{./unitary.tex}
 \input{./sa.tex}
 \input{./order.tex}
+\input{./homomorphism.tex}
+\input{./gelfand.tex}

src/topology/main/lch.tex

@@ -159,7 +159,7 @@
 \end{proof}
 
 \begin{lemma}
-\label{lemma:lch-locally-finite-relatively compact-refine}
+\label{lemma:lch-locally-finite-relatively-compact-refine}
     Let $X$ be a LCH space and $\ce \subset 2^X$ be a locally finite relatively compact open cover of $X$, then there exists locally finite relatively compact open covers $\bracs{F_E}_{E \in \ce}, \bracs{G_E}_{E \in \ce} \subset 2^X$ such that for each $E \in \ce$, $G_E \subset \ol{G_E} \subset E \subset \ol{E} \subset F_E$.
 \end{lemma}
 \begin{proof}
@@ -223,7 +223,7 @@
 \begin{proof}
     (1) $\Rightarrow$ (2): For each $x \in X$, there exists a relatively compact open neighbourhood $U_x \in \cn^o(x)$. Since $\bracs{U_x| x \in X}$ is an open cover of $X$, there exists a locally finite refinement $\mathcal{V}$. For each $V \in \mathcal{V}$, there exists $x \in X$ such that $V \subset U_x$. In which case, $\ol{V} \subset \ol{U_x}$ is compact.
 
-    (2) $\Rightarrow$ (3): Let $\cf \subset 2^X$ be a locally finite open cover of $X$ consisting of relatively compact open sets. By \autoref{lemma:lch-locally-finite-relatively compact-refine}, there exists a locally finite open cover $\bracs{G_F}_{F \in \cf}$ of $X$ consisting of relatively compact open sets such that $\ol{F} \subset G_F$ for all $F \in \cf$.
+    (2) $\Rightarrow$ (3): Let $\cf \subset 2^X$ be a locally finite open cover of $X$ consisting of relatively compact open sets. By \autoref{lemma:lch-locally-finite-relatively-compact-refine}, there exists a locally finite open cover $\bracs{G_F}_{F \in \cf}$ of $X$ consisting of relatively compact open sets such that $\ol{F} \subset G_F$ for all $F \in \cf$.
 
     For each $F \in \cf$, let
     \[
@@ -239,7 +239,7 @@
 
     is finite, and $\mathcal{V}$ is locally finite.
 
-    (3) $\Rightarrow$ (4): By \autoref{lemma:lch-locally-finite-relatively compact-refine}.
+    (3) $\Rightarrow$ (4): By \autoref{lemma:lch-locally-finite-relatively-compact-refine}.
 
     (4) $\Rightarrow$ (5): Let $\seqi{V}, \seqi{W} \subset 2^X$ be locally finite refinements of $\mathcal{U}$ consisting of relatively compact open sets such that for each $i \in I$, $\ol{W_i} \subset V_i$.