Review.

.vscode/project.code-snippets

@@ -181,5 +181,10 @@
     "scope": "latex",
     "prefix": "scaf", 
     "body": ["\\hyperref[scaffolded]{definition:measure-scaffold}$0"]
-  }
+  },
+  "Locally Compact Abelian Group": {
+  "scope": "latex",
+    "prefix": "lcag",
+    "body": ["locally compact Abelian group$0"]
+  },
 }

refs.bib

@@ -308,4 +308,15 @@
   volume={16},
   year={2018},
   publisher={Edizioni della normale Pisa}
+}
+
+@article{GleasonProjective,
+  title={Projective topological spaces},
+  author={Gleason, Andrew M},
+  journal={Illinois Journal of Mathematics},
+  volume={2},
+  number={4A},
+  pages={482--489},
+  year={1958},
+  publisher={Duke University Press}
 }

src/fa/lc/compact.tex

@@ -24,7 +24,7 @@
 
 \begin{definition}[Extreme Point]
 \label{definition:extreme-point}
-    Let $E$ be a vector space over $\real$, $K \subset E$, and $x \in K$, then $x$ is \textbf{extremal} if there exists no $y, z \in K$ such that $x \in (y, z) \subset K$. 
+    Let $E$ be a vector space over $\real$, $K \subset E$, and $x \in K$, then $x$ is \textbf{extremal} if there exists no $y, z \in K \setminus \bracsn{x}$ such that $x \in (y, z) \subset K$. 
 \end{definition}
 
 \begin{definition}[Extreme Subset]
@@ -32,6 +32,13 @@
     Let $E$ be a vector space over $\real$, $K \subset E$ be convex, and $A \subset K$, then $A$ is \textbf{extreme set} if for any $x \in A$ and $y, z \in K$ such that $x \in (y, z)$, $y, z \in A$ as well.  
 \end{definition}
 
+\begin{lemma}
+\label{lemma:extreme-subset-extreme-point}
+    Let $E$ be a vector space over $\real$, $K \subset E$ be convex, $A \subset K$ be extreme, and $x \in A$ be an extreme point of $A$, then $x$ is an extreme point of $K$. 
+\end{lemma}
+\begin{proof}
+    Let $y, z \in K$ such that $x \in (y, z)$. Since $A$ is extreme, $y, z \in A$. As $x$ is extremal in $A$, $x = y = z$, so $x$ is extremal in $K$. 
+\end{proof}
 
 \begin{lemma}
 \label{lemma:extremal-face}
@@ -135,11 +142,3 @@
 
     Now, let $A'$ be the closed convex hull of $\seq{x_n}$ with respect to $E_B$. Since the inclusion $E_B \to E$ is continuous, $A'$ is a compact convex set in $E$ by \autoref{proposition:compact-extensions}. As such, $A' = A$ by \autoref{proposition:closure-of-image} and \autoref{proposition:compact-closed}. Therefore $A$ is a compact subset of $E_B$. 
 \end{proof}
-
-\begin{lemma}
-\label{lemma:auxiliary-weak-dense}
-    Let $E$ be a separated locally convex space over $K \in \RC$, $A \subset E$ be compact, convex, and circled, and $
-\end{lemma}
-
-
-

src/fa/tvs/complete-metric.tex

@@ -150,3 +150,13 @@
 
 \end{proof}
 
+
+\begin{example}
+    \label{example:open-mapping-counters}
+
+    Let $E = l^1(\natp; \complex)$ and $F$ be $l^1(\natp; \complex)$ equipped with the uniform norm, then the identity $E \to F$ is continuous but not open. 
+
+    Let $E = C^1([0, 1]; \complex)$ equipped with the uniform norm, $F = C([0, 1]; \complex)$, and $D: E \to F$ be the differentiation operator, then $D$ has closed graph, but is not continuous. 
+\end{example}
+
+

src/harmonic/commutative/dual.tex

@@ -0,0 +1,106 @@
+\section{Pontryagin Duality}
+\label{section:lacg-dual}
+
+
+\begin{definition}[Unitary Character]
+\label{definition:unitary-character}
+    Let $G$ be a locally compact Abelian group and $\phi: G \to \mathbb{T}$, then $\phi$ is a \textbf{unitary character} of $G$ if $\phi$ is a continuous homomorphism. 
+
+    The set $\wh G$ of all unitary characters of $G$, equipped with pointwise operations and the topology of compact convergence, is the \textbf{dual group} of $G$. 
+
+    For any $\phi \in \wh G$ and $x \in G$, denote $\phi(x) = \dpn{x, \phi}{G}$. 
+\end{definition}
+
+\begin{theorem}
+\label{theorem:unitary-character-isomorphism}
+    Let $G$ be a locally compact Abelian group and $\phi \in \wh G$. Under the identification that $\mathbb{T} = \partial B_\complex(0, 1)$ and $B(\complex) = \complex$, the representation of $L^1(G; \complex)$ determined by $\phi$ takes the form
+    \[
+    \pi_\phi: L^1(G; \complex) \to \complex \quad \dpn{f, \pi_\phi}{L^1(G; \complex)} = \int_G f(x)\dpn{x, \phi}{G} dx
+    \]
+
+    and the mapping $\wh G \to \Omega(L^1(G; \complex))$ defined by $\phi \mapsto \pi_\phi$ is a homeomorphism. Moreover, for any $f \in L^1(G; \complex)$ with $\pi_\phi(f) \ne 0$, $\dpn{x, \phi}{G} = \pi_\phi(L_xf)/\pi_\phi(f)$ for all $x \in G$. 
+\end{theorem}
+\begin{proof}[Proof, {{\cite[Theorem 4.3]{FollandHarmonic}}}. ]
+    (Continuous): Let $\seqf{f_j} \subset L^1(G; \complex)$ and $\eps > 0$. By inner regularity of the Haar measure, there exists $K \subset G$ compact such that $\int_{G\setminus K}|f_j| < \eps$ for all $1 \le j \le n$. In which case, for any $\phi, \psi \in \wh G$ with $\sup_{x \in K}|\phi(x) - \psi(x)| < \eps$, 
+    \begin{align}
+    \abs{\dpn{f_j, \phi - \psi}{L^1(G; \complex)}} &\le \int_{K}|f_j \cdot (\phi - \psi)| + 2\int_{G \setminus K}|f_j| \\
+    &\le \eps \norm{f_j}_{L^1(G; \complex)} + 2\eps 
+    \end{align}
+
+    (Injective): By (3) of \autoref{definition:representation-l1-group-algebra}, $\phi$ is uniquely determined by $\pi_\phi$. 
+
+    (Surjective): Let $\pi \in \Omega(L^1(G; \complex))$. Since $G$ is Abelian, $L^1(G; \complex)$ is also Abelian by \autoref{proposition:convolution-measures-property}. As such, for any $x \in G$ and $f, g \in L^1(G; \complex)$, $f * (L_xg) = L_x(f * g) = (L_xf) * g$ and $\pi(L_xf)\pi(g) = \pi(f)\pi(L_xg)$. In particular, if $\pi(f), \pi(g) \ne 0$, then $\pi(L_xf)/\pi(f) = \pi(L_xg)/\pi(g)$. Let\footnote{It may be tempting to construct $\phi$ directly as an element of $L^\infty$. However, this requires either reducing to the $\sigma$-compact case, or expanding the measurable sets of the Haar measure.}
+    \[
+    \phi: G \to \complex \quad x \mapsto \frac{\pi(L_xf)}{\pi(f)}
+    \]
+
+    then $\phi$ is defined independently of the choice of $f$. By \hyperref[continuity of translation]{proposition:haar-translation}, $\phi$ is a continuous homomorphism from $G$ to $\complex \setminus \bracsn{0}$. Moreover, by \autoref{proposition:multiplicative-less-unit}, $\norm{\pi}_{L^1(G; \complex)^*} \le 1$, and $\phi$ is bounded. Given that $\phi$ is a homomorphism, $\phi(G) \subset \partial B_\complex(0, 1)$, and $\phi \in \wh G$. 
+
+    Since $\pi \ne 0$, there exists $\psi \in L^1(G; \complex)$ with $\pi(\psi) = 1$. Using \hyperref[Riemann sums]{lemma:convolution-distribute-functional}, 
+    \begin{align*}
+        \pi(f) &= \pi(f)\pi(\psi) = \pi(f * \psi) = \int_G f(y)\pi(L_y\psi) dy \\
+        &= \int_G f(y)\frac{\pi(L_y\psi)}{\pi(\psi)}dy = \int_G f(y) \dpn{y, \phi}{G} dy
+    \end{align*}
+
+    (Embedding): By \autoref{proposition:multiplicative-less-unit}, every multiplicative linear functional on $L^1(G; \complex)$ is contractive. Hence $\Omega(L^1(G; \complex))$ is equicontinuous. By the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, the topology of compact convergence on $\Omega(L^1(G; \complex))$ coincides with the weak* topology.  
+
+    Let $\phi, \psi \in \wh G$. Since $\phi \ne 0$, there exists $f \in L^1(G; \complex)$ such that $\pi_\phi(f) \ne 0$. If $|\pi_\phi(f) - \pi_\psi(f)|$ is sufficiently small, then $\pi_\psi(f) \ne 0$ as well. Let $K \subset G$ be compact, then $\bracsn{L_xf|x \in K}$ is compact in $L^1(G; \complex)$ by \hyperref[continuity of translation]{proposition:haar-translation}. In which case, 
+    \[
+    \sup_{x \in K}|(\phi - \psi)(x)| = \sup_{x \in K}\abs{\frac{\pi_\phi(L_xf)}{\pi_\phi(f)} - \frac{\pi_\psi(L_xf)}{\pi_\psi(f)}}
+    \]
+    
+    Thus for any $\eps > 0$, $\bracsn{\pi_\psi|\psi \in \wh G, \sup_{x \in K}|\phi(x) - \psi(x)| \le \eps}$ is a neighbourhood of $\pi_\phi$ in the weak* topology, and the mapping $\psi \mapsto \pi_\psi$ is an embedding. 
+\end{proof}
+
+\begin{proposition}
+\label{proposition:unitary-character-orthonormal}
+    Let $G$ be a compact Abelian group with $|G| = 1$, then $\wh G$ is an orthonormal set in $L^2(G; \complex)$. 
+\end{proposition}
+\begin{proof}[Proof, {{\cite[Proposition 4.4]{FollandHarmonic}}}. ]
+    Let $\phi, \psi \in \wh G$ be distinct, then there exists $x_0 \in G$ with $\dpn{x_0, \phi}{G}\dpn{x_0, \psi}{G}^{-1} \ne 1$. In which case, 
+    \begin{align*}
+        \int_G \phi \ol{\psi} &= \int_G \dpn{x, \phi}{G}\dpn{x, \psi}{G}^{-1} dx \\
+        &= \dpn{x_0, \phi}{G}\dpn{x_0, \psi}{G}\int_G \dpn{x - x_0, \phi}{G}\dpn{x - x_0, \psi}{G} dx \\
+        &= \dpn{x_0, \phi}{G}\dpn{x_0, \psi}{G} \int \phi \ol{\psi}
+    \end{align*}
+
+    so $\phi \perp \psi$. 
+\end{proof}
+
+\begin{proposition}
+\label{proposition:pontryagin-compact-discrete}
+    Let $G$ be a locally compact Abelian group. If $G$ is discrete, then $\wh G$ is compact. If $G$ is compact, then $\wh G$ is discrete. 
+\end{proposition}
+\begin{proof}[Proof, {{\cite[Proposition 4.5]{FollandHarmonic}}}. ]
+    If $G$ is discrete, then $L^1(G; \complex)$ is a unital Banach algebra. By \autoref{definition:multiplicative-linear-functional-space}, $\wh G \iso \Omega(L^1(G; \complex))$ is compact. 
+
+    If $G$ is compact, then $\one \in L^1(G; \complex)$, and $\bracsn{f \in L^\infty(G; \complex)|\int f| > 1/2}$ is weak*-open. By \autoref{proposition:unitary-character-orthonormal}, for any $\phi \in \wh G \setminus \bracsn{\one}$, $\int \phi = 0$. Therefore $\bracsn{1}$ is open in $\wh G$, and $\wh G$ is discrete. 
+\end{proof}
+
+\begin{proposition}
+\label{proposition:pontryagin-product}
+    Let $\seqi{G}$ be locally compact Abelian groups where $G_i$ is compact for all but finitely many $i \in I$, then
+    \[
+    \braks{\prod_{i \in I}G_i}^\wedge \iso \bigoplus_{i \in I}\wh G_i
+    \]
+\end{proposition}
+
+\begin{theorem}
+\label{theorem:pontryagin-dual-usual-suspects}
+    \begin{enumerate}
+        \item $\wh \real \iso \real$ with dual pairing $\dpn{x, \xi}{\real} = e^{2\pi i \xi x}$. 
+        \item $\wh \mathbb{T} \iso \integer$ with dual pairing $\dpn{x, n}{\mathbb{T}} = x^n$. 
+        \item $\wh \integer = \wh \mathbb{T}$ with dual pairing $\dpn{n, x}{\mathbb{T}} = x^n$. 
+    \end{enumerate}
+\end{theorem}
+\begin{proof}[Proof, {{\cite[Proposition 4.6]{FollandHarmonic}}}. ]
+    (1): Let $\phi \in \wh \real$, then $\dpn{0, \phi}{\real} = 1$. Thus there exists $\alpha > 0$ with $A := \int_0^\alpha \dpn{x, \phi}{\real}dx \ne 0$. In which case, 
+    \[
+    A \dpn{x, \phi}{G} = \int_0^\alpha \dpn{x + y, \phi}{\real} dy = \int_x^{x + \alpha} \dpn{y, \phi}{\real}dy
+    \]
+
+    so $\phi \in C^1(\real)$ and there exists $c \in \complex$ with $\phi' = c\phi$. Therefore $\dpn{x, \phi}{\real} = e^{cx}$ for all $x \in \real$. 
+\end{proof}
+
+
+

src/harmonic/commutative/index.tex

@@ -0,0 +1,4 @@
+\chapter{Locally Compact Abelian Groups}
+\label{chap:lcag}
+
+\input{./dual.tex}

src/harmonic/lcg/algebra.tex

@@ -306,6 +306,64 @@ Unfortunately, $L^1(G)$ is not a $C^*$-algebra, and its approximate identity req
     and $f * \phi_U \to f$ in $L^p(G; \complex)$ by \hyperref[continuity of translation]{proposition:haar-translation}.
 \end{proof}
 
+
+
+\begin{lemma}[Riemann Sums]
+\label{lemma:convolution-distribute-functional}
+    Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$, then for any $f, g \in L^1(G; \complex)$, 
+    \[
+    \dpn{f * g, \phi}{L^1(G; \complex)} = \int_G f(y)\dpn{L_yg, \phi}{L^1(G; \complex)} dy
+    \]
+\end{lemma}
+\begin{proof}
+    Let $K \subset G$ be compact and $\eps > 0$. By continuity of translation, there exists a partition $K = \bigsqcup_{j = 1}^n U_j$ of $K$ and $\seqf{y_j}$ such that:
+    \begin{enumerate}
+        \item For each $1 \le j \le n$, $y_j \in U_j$. 
+        \item For each $1 \le j \le n$ and $y \in U_j$,
+        \[
+        \normn{L_yg - L_{y_j}g}_{L^1(G; \complex)} \vee \normn{\dpn{L_yg, \phi}{L^1(G; \complex)} - \dpn{L_{y_j}g, \phi}{L^1(G; \complex)}} < \eps
+        \]
+    \end{enumerate}
+    
+    Since $g \in L^1(G; \complex)$, $\bracsn{g \ne 0}$ is $\sigma$-finite. By \hyperref[Tonelli's Theorem]{theorem:fubini-tonelli}, 
+    \begin{align*}
+        \norm{\one_K * g - \sum_{j = 1}^n|U_j| L_{y_j}g}_{L^1(G)} &= \int_G\abs{\one_K * g(x) - \sum_{j = 1}^n|U_j| L_{y_j}g(x)}dx \\
+        &\le \iint_{G \times G}\abs{\one_K(y)L_yg(x) - \sum_{j = 1}^n\one_{U_j}(y) L_{y_j}g(x)} dy dx \\
+        &\le \sum_{j = 1}^n \int_G\int_{U_j}|L_yg(x) - L_{y_j}g(x)|dydx \\
+        &\le \sum_{j = 1}^n \eps |U_j| = \eps |K|
+    \end{align*}
+
+    Similarly, 
+    \[
+    \abs{\int_G \one_K(y)\dpn{L_yg, \phi}{L^1(G; \complex)}dy - \sum_{j = 1}^n |U_j|\dpn{L_{y_j}g, \phi}{L^1(G; \complex)}} \le \eps |K|
+    \]
+
+    By linearity, 
+    \[
+    \angles{\sum_{j = 1}^n |U_j|L_{y_j}g, \phi}_{L^1(G; \complex)} = \sum_{j = 1}^n |U_j| \dpn{L_{y_j}g, \phi}{L^1(G; \complex)}
+    \]
+
+    so
+    \[
+    \abs{\dpn{\one_K * g, \phi}{L^1(G; \complex)} - \int \one_K(y)\dpn{L_y g, \phi}{L^1(G; \complex)}dy} \le 2\eps |K| (1 \vee \norm{\phi}_{L^1(G; \complex)^*})
+    \]
+
+    As the above holds for all $\eps > 0$, $\dpn{\one_K * g, \phi}{L^1(G; \complex)} = \int_G\one_K(y)\dpn{L_y g, \phi}{L^1(G; \complex)}dy$. 
+
+    By inner regularity of the Haar measure,
+    \[
+    \dpn{\one_B * g, \phi}{L^1(G; \complex)} = \int_G\one_B(y)\dpn{L_y g, \phi}{L^1(G; \complex)}dy
+    \]
+    
+    for all $B \in \cb_\complex$. By \hyperref[density of simple functions]{proposition:lp-simple-dense}, 
+    \[
+    \dpn{f * g, \phi}{L^1(G; \complex)} = \int_G f(y)\dpn{L_y g, \phi}{L^1(G; \complex)} dy
+    \]
+
+    for all $f \in L^1(G; \complex)$. 
+    
+\end{proof}
+
 \begin{theorem}
 \label{theorem:lcg-group-algebra-ideal}
     Let $G$ be a locally compact group and $J \subset L^1(G; \complex)$ be a closed subspace, then $J$ is a left ideal if and only if $L_xJ \subset J$ for all $x \in G$. 
@@ -331,3 +389,4 @@ Unfortunately, $L^1(G)$ is not a $C^*$-algebra, and its approximate identity req
 \end{proof}
 
 
+

src/harmonic/notation.tex

@@ -1,3 +1,41 @@
 \chapter{Notations}
 \label{chap:harmonic-notations}
 
+\begin{tabular}{lll}
+    \textbf{Notation} & \textbf{Description} & \textbf{Source} \\
+    \hline
+
+    % ---- Locally Compact Groups ----
+    $1 = 1_G$ & Identity element of the group $G$. & \autoref{lemma:lc-sigma-compact} \\
+    $C_c^+(G)$ & Space of non-negative continuous compactly supported functions on $G$. & \autoref{definition:lcg-covering-ratio} \\
+    $L_x f$ & Left translation of $f$ by $x$, $L_xf(y) = f(x^{-1}y)$. & \autoref{definition:lcg-covering-ratio} \\
+    $R_x f$ & Right translation of $f$ by $x$, $R_xf(y) = f(yx)$. & \autoref{definition:modular-function} \\
+    % ---- Haar Measures ----
+    $(f : g)$ & Covering ratio of $f$ by $g$. & \autoref{definition:lcg-covering-ratio} \\
+    $L^p(G; E)$ & $L^p(G, \mu; E)$, where $\mu$ is a fixed left Haar measure on $G$. & \autoref{definition:haar-measure-lp-notation} \\
+    $\int_G f(x) dx$ & Integral of $f$ with respect to the fixed left Haar measure. & \autoref{definition:haar-measure-lp-notation} \\
+    $|A|$ & Haar measure of a Borel set $A \subset G$. & \autoref{definition:haar-measure-lp-notation} \\
+    % ---- Modular Function ----
+    $\Delta_G$ & Modular function of $G$. & \autoref{definition:modular-function} \\
+    % ---- Group Algebras ----
+    $\mu * \nu$ & Convolution of finite Radon measures. & \autoref{definition:lcg-convolution-measures} \\
+    $\mu^*$ & Involution of a measure, $\mu^*(A) = \overline{\mu(A^{-1})}$. & \autoref{definition:group-measure-algebra} \\
+    $f * g$ & Convolution of functions on $G$. & \autoref{definition:group-convolution-functions} \\
+    $f^*$ & Involution on $L^1(G; \complex)$, $f^*(x) = \Delta_G(x^{-1})\overline{f(x^{-1})}$. & \autoref{definition:l1-group-algebra} \\
+    % ---- Representations ----
+    $\bigoplus_{i \in I}\pi_i$ & Direct sum of unitary representations. & \autoref{definition:lcg-rep-direct-sum} \\
+    $\pi_L$ & Left regular representation of $G$. & \autoref{definition:lcg-left-regular} \\
+    $\mathcal{C}(\pi_1, \pi_2)$ & Space of intertwining operators between $\pi_1$ and $\pi_2$. & \autoref{definition:lcg-intertwining} \\
+    % ---- GNS Construction ----
+    $S(G)$ & State space of $G$. & \autoref{definition:group-l1-state}, \autoref{definition:group-positive-definite-function} \\
+    $\langle f, g \rangle_\phi$ & Pseudo inner product $\langle g^* * f, \phi \rangle_{L^1(G; \complex)}$ induced by a positive linear functional $\phi$. & \autoref{definition:gns-triple-lcg} \\
+    $(H_\phi, \pi_\phi, \xi_\phi)$ & GNS triple associated with $\phi$. & \autoref{definition:gns-triple-lcg} \\
+    % ---- Positive-Definite Functions ----
+    $P(G)$ & Space of positive-definite functions on $G$. & \autoref{definition:group-positive-definite-function} \\
+    $P_0(G)$ & Space of positive-definite functions of norm at most $1$. & \autoref{definition:group-positive-definite-function} \\
+    % ---- Locally Compact Abelian Groups ----
+    $\mathbb{T}$ & Unit circle $\partial B_\complex(0, 1)$. & \autoref{definition:unitary-character} \\
+    $\widehat{G}$ & Dual group of a locally compact Abelian group $G$. & \autoref{definition:unitary-character} \\
+    $\langle x, \phi \rangle_G$ & Dual pairing, $\langle x, \phi \rangle_G = \phi(x)$ for $\phi \in \widehat{G}$. & \autoref{definition:unitary-character} \\
+    $\pi_\phi$ & Representation of $L^1(G; \complex)$ determined by the character $\phi \in \widehat{G}$. & \autoref{theorem:unitary-character-isomorphism}
+\end{tabular}

src/harmonic/rep/gns.tex

@@ -0,0 +1,182 @@
+\section{The GNS Construction for Groups}
+\label{section:gns-lcg}
+
+\begin{definition}[Positive Linear Functional]
+\label{definition:group-l1-positive-functional}
+    Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$\footnote{Due to technical difficulties, positive linear functionals are a priori not represented by $L^\infty$ functions. }, then $\phi$ is \textbf{positive} if $\dpn{f^* * f,  \phi}{L^1(G; \complex)} \ge 0$ for all $f \in L^1(G)$. 
+\end{definition}
+
+\begin{definition}[State]
+\label{definition:group-l1-state}
+    Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$ be a positive linear functional, then $\phi$ is a \textbf{state} if $\norm{\phi}_{L^1(G; \complex)^*} = 1$. The set $S(G)$ of all states on $L^1(G; \complex)$ is the \textbf{state space} of $G$. 
+\end{definition}
+
+\begin{definition}[GNS Triple (Locally Compact Group)]
+\label{definition:gns-triple-lcg}
+    Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$ be a positive linear functional. For each $f, g \in L^1(G; \complex)$, let
+    \[
+    \dpn{f, g}{\phi} = \dpn{g^* * f, \phi}{L^1(G; \complex)}
+    \]
+
+    Let $N_\phi = \bracsn{f \in L^1(G; \complex)| \dpn{f, f}{\phi} = 0}$, $H_\phi^0 = L^1(G; \complex)/N_\phi$, $H_\phi$ be its completion with respect to $\dpn{\cdot, \cdot}{\phi}$, and
+    \[
+    \pi_\phi^0: G \to B(H_\phi^0) \quad \pi_\phi^0(x)(f + N_\phi) = L_xf + N_\phi
+    \]
+
+    For each $x \in G$, let $\pi_\phi(x)$ be the continuous extension of $\pi_\phi^0(x)$ to an element of $B(H_\phi)$, then:
+    \begin{enumerate}
+        \item $H_\phi$ equipped with the continuous extension of $\dpn{\cdot, \cdot}{\phi}$ is a Hilbert space.
+        \item $(H_\phi, \pi_\phi)$ is a well-defined unitary representation of $G$. 
+    \end{enumerate}
+
+    Let $\pi_\phi: L^1(G; \complex) \to B(H_\phi)$ also denote the *-representation of $L^1(G; \complex)$ determined by $\pi_\phi$, then:
+    \begin{enumerate}[start=2]
+        \item For each $f, g \in L^1(G; \complex)$, $\pi_\phi(f)(g + N_\phi) = f * g + N_\phi$. 
+        \item There exists a cyclic vector $\xi_\phi \in H_\phi$ for $\pi_\phi$ such that for each $f, g \in L^1(G; \complex)$, 
+        \[
+        \dpn{f, g}{\phi} = \dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi}
+        \]
+        \item For each $x \in G$, let $\Phi(x) = \dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}$, then $\Phi \in BC(G; \complex)$ and
+        \[
+        \dpn{f, \phi}{L^1(G; \complex)} = \int_G f(x)\Phi(x)dx
+        \]
+
+        for all $f \in L^1(G; \complex)$. 
+    \end{enumerate}
+
+    Moreover, 
+    \begin{enumerate}
+        \item[(U)] Let $(K, \tau)$ be a unitary representation of $G$ with cyclic vector $\eta$ such that $\dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi} = \dpn{\tau(x)\eta, \eta}{K}$, then there exists a unitary equivalence $U: H_\phi \to K$ such that $U\xi_\phi = \eta$. 
+    \end{enumerate}
+    
+
+    The representation $(H_\phi, \pi_\phi)$ is the \textbf{unitary cyclic representation of $G$ induced by $\phi$}, and the triple $(H_\phi, \pi_\phi, \xi_\phi)$ is the \textbf{Gelfand-Naimark-Segal (GNS) triple associated with $\phi$}. 
+\end{definition}
+\begin{proof}[Proof, {{\cite[Theorem 3.20]{FollandHarmonic}}}. ]
+    (1): Since $\phi$ is positive, $\dpn{\cdot, \cdot}{\phi}$ is a pseudo inner product, and $H_\phi$ equipped with its continuous extension is a Hilbert space. 
+    
+    (2): By the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}, for any $f, g \in L^1(G; \complex)$, 
+    \[
+    |\dpn{f, g}{\phi}| \le \dpn{f, f}{\phi}^{1/2} \cdot \dpn{g, g}{\phi}^{1/2}
+    \]
+
+    In particular, if $g \in N_\phi$, then
+    \begin{align*}
+    \dpn{f * g, f * g}{\phi} &= \dpn{g^* * f^* * f * g, \phi}{L^1(G; \complex)} \\
+    &= \dpn{g^** (f^**f*g), \phi}{L^1(G; \complex)} = \dpn{f^**f*g, g}{\phi}
+    \end{align*}
+    
+
+    so $N_\phi$ is a left ideal of $L^1(G; \complex)$. Let $x \in G$, then $(L_xg)^* = L_{-x}g^*$, and
+    \begin{align*}
+        \dpn{L_xf, L_xg}{\phi}  &= \dpn{(L_xg)^* * L_xf, \phi}{L^1(G; \complex)} = \dpn{(L_{-x}g^*) * (L_xf), \phi}{L^1(G; \complex)} \\
+        &= \dpn{g^* * f, \phi}{L^1(G; \complex)} = \dpn{f, g}{\phi}
+    \end{align*}
+
+    by \autoref{proposition:convolution-function-gymnastics}. Therefore $\pi_\phi^0(x)$ extends to a unitary map on $H_\phi$, and $\pi_\phi$ is a well-defined unitary representation of $G$, which is strong-operator continuous by \hyperref[continuity of translation]{proposition:haar-translation}.
+
+    (3): Let $f, g, h \in L^1(G; \complex)$, then after passing through \hyperref[Riemann sums]{lemma:convolution-distribute-functional}, 
+    \begin{align*}
+        \dpn{\pi_\phi(f)g + N_\phi, h + N_\phi}{H_\phi} &= \int_G f(x)\dpn{\pi_\phi(x)g + N_\phi, h + N_\phi}{H_\phi} dx \\
+        &= \int_G f(x)\dpn{L_xg, h}{\phi} dx = \dpn{f * g, h}{\phi} \\
+        &= \dpn{f * g + N_\phi, h + N_\phi}{H_\phi}
+    \end{align*}
+    
+    As the above holds for all $h \in L^1(G; \complex)$, $\pi_\phi(f)(g + N_\phi) = f * g + N_\phi$. 
+
+    (4): Using \autoref{proposition:group-algebra-approximate-identity}, let $\angles{\psi_\alpha}_{\alpha \in A} \subset L^1(G; \complex)$ be an approximate identity, and $\xi_\phi \in H_\phi$ be defined by
+    \[
+    \dpn{\xi_\phi, g + N_\phi}{H_\phi} = \lim_{\alpha \in A}\dpn{\psi_\alpha, g}{\phi} \quad \forall g \in L^1(G; \complex)
+    \]
+
+    then $\normn{\xi_\phi}_{H_\phi} \le \norm{\phi}_{L^1(G; \complex)^*}^{1/2}$. Moreover, for any $f, g \in L^1(G; \complex)$, 
+    \begin{align*}
+        \dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} &= \lim_{\alpha \in A}\dpn{\pi_\phi(f)(\psi_\alpha + N_\phi), \pi_\phi(g)\xi_\phi}{H_\phi} \\
+        &= \lim_{\alpha \in A}\dpn{f * \psi_\alpha + N_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} \\
+        &= \dpn{f + N_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} = \dpn{f + N_\phi, g + N_\phi}{H_\phi}
+    \end{align*}
+\end{proof}
+
+\begin{corollary}
+\label{corollary:group-l1-positive-function}
+    Let $G$ be a locally compact group, $\phi \in L^1(G; \complex)^*$ be a positive linear functional, $(H_\phi, \pi_\phi, \xi_\phi)$ be the GNS triple associated with $\phi$. Then:
+    \begin{enumerate}
+        \item For each $x \in G$, let $\Phi(x) = \dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}$, then $\Phi \in BC(G; \complex)$ with
+        \[
+        \dpn{f, \phi}{L^1(G; \complex)} = \int_G f(x)\Phi(x) dx
+        \]
+
+        for all $f \in L^1(G; \complex)$. 
+        \item $\Phi(1_G) = \norm{\Phi}_u = \norm{\phi}_{L^1(G; \complex)^*}$.
+    \end{enumerate}
+\end{corollary}
+\begin{proof}
+    (1): For any $f \in L^1(G; \complex)$
+    \[
+    \dpn{\pi_\phi(f)\xi_\phi, \xi_\phi}{H_\phi} = \int_G f(x)\dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}dx = \int_G f(x)\Phi(x)dx
+    \]
+
+    by definition of the representation. Since $\pi_\phi$ is strong-operator continuous, $\Phi \in C(G; \complex)$. 
+    
+    (2): By \autoref{theorem:lp-dual-function}, $\Phi \in BC(G; \complex)$ with $\norm{\Phi}_u = \norm{\phi}_{L^1(G; \complex)^*}$. In addition, for any $x \in G$, 
+    \[
+    |\Phi(x)| = |\dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}| \le \normn{\xi_\phi}_{H_\phi}^2 = \Phi(1_G)
+    \]
+\end{proof}
+
+
+\begin{proposition}
+\label{proposition:gns-irreducible-lcg}
+    Let $G$ be a locally compact group, $\phi \in S(G)$ be a state, and $(H_\phi, \pi_\phi, \xi_\phi)$ be the GNS triple associated with $\phi$, then $(H_\phi, \pi_\phi)$ is irreducible if and only if $\phi$ is an extreme point of $S(G)$. 
+\end{proposition}
+\begin{proof}
+    ($\Rightarrow$): Suppose that $(H_\phi, \pi_\phi)$ is irreducible. Let $\psi_1, \psi_2 \in S(G)$ and $t \in (0, 1)$ such that $\phi = t\psi_1 + (1 - t)\psi_2$. Define
+    \[
+    T: H_\phi \to H_\phi \quad \dpn{T\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} = \dpn{f, g}{\psi_1} = \dpn{g^* * f, \psi_1}{L^1(G; \complex)}
+    \]
+
+    where $f, g \in L^1(G; \complex)$. Since both $\psi_1$ and $\psi_2$ are positive-definite functions, $\phi - t\psi_1$ is also positive-definite. In particular, $\dpn{f, f}{\psi_1} \le t^{-1}\dpn{f, f}{\phi}$ for all $f \in L^1(G; \complex)$, so $T$ extends continuously into a positive operator on $H_\phi$. 
+
+    For any $x \in G$ and $f, g \in L^1(G; \complex)$, 
+    \begin{align*}
+        \dpn{\pi_\phi(x)T\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} &= \dpn{T\pi_\phi(f)\xi_\phi, \pi_\phi(x^{-1})\pi_\phi(g)\xi_\phi}{H_\phi} \\
+        &= \dpn{T\pi_\phi(f)\xi_\phi, \pi_\phi(L_{x^{-1}}g)\xi_\phi}{H_\phi} \\
+        &= \dpn{f, L_{x^{-1}}g}{\psi_1} = \dpn{(L_{x^{-1}}g)^* * f, \psi_1}{L^1(G; \complex)} \\
+        &= \dpn{g^* * (L_xf), \psi_1}{L^1(G; \complex)} = \dpn{L_xf, g}{\psi_1} \\
+        &= \dpn{T\pi_\phi(L_xf)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} \\
+        &= \dpn{T\pi_\phi(x)\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi}
+    \end{align*}
+
+    so $T \in \pi_\phi(G)'$. By \hyperref[Schur's Lemma]{lemma:lcg-schur}, there exists $\lambda \in \complex \setminus \bracsn{0}$ such that $T = \lambda I$. 
+
+    Using \autoref{proposition:group-algebra-approximate-identity}, let $\angles{g_\alpha}_{\alpha \in A} \subset L^1(G; \complex)$ be an approximate identity for $L^1(G; \complex)$, then for any $f \in L^1(G; \complex)$, 
+    \[
+    \dpn{f, \psi_1}{L^1(G; \complex)} = \lim_{\alpha \in A}\dpn{f, g_\alpha}{\psi_1} = \lim_{\alpha \in A}\lambda\dpn{f, g_\alpha}{\phi} = \lambda \dpn{f, \phi}{L^1(G; \complex)}
+    \]
+
+    Since $\phi, \psi_1 \in S(G)$, $\lambda = 1$, and $\phi = \psi_1$. As the argument is symmetric, $\phi = \psi_1 = \psi_2$. 
+
+    ($\Leftarrow$): Suppose that $(H_\phi, \pi_\phi)$ is reducible. By \hyperref[Schur's Lemma]{lemma:lcg-schur}, $\pi_\phi(G)'$ is non-trivial. Since $\pi_\phi(G)'$ is a von Neumann algebra, it admits a non-trivial projection $P \in B(H_\phi)$ by \autoref{theorem:vn-projection-norm-dense}. 
+    
+    Let $\xi_1 = P\xi_\phi/\normn{P\xi_\phi}_{H_\phi}$ and $\xi_2 = (1 - P)\xi_\phi/\normn{\xi_\phi - P\xi_\phi}_{H_\phi}$. For each $f \in L^1(G; \complex)$, let
+    \[
+    \dpn{f, \psi_1}{L^1(G; \complex)} = \dpn{\pi_\phi(f)\xi_1, \xi_1}{H_\phi} \quad
+    \dpn{f, \psi_2}{L^1(G; \complex)} = \dpn{\pi_\phi(f)\xi_2, \xi_2}{H_\phi}
+    \]
+
+    then $\psi_1, \psi_2 \in S(G)$. Since $P \in \pi_\phi(G)'$, for any $f \in L^1(G; \complex)$, 
+    \begin{align*}
+        \dpn{f, \phi}{L^1(G; \complex)} &= \dpn{\pi_\phi(f)\xi_\phi, \xi_\phi}{H_\phi} \\
+        &= \dpn{\pi_\phi(f)P\xi_\phi, P\xi_\phi}{H_\phi} + \dpn{\pi_\phi(f)(I - P)\xi_\phi, (I - P)\xi_\phi}{H_\phi} \\
+        &= \normn{P\xi_\phi}_{H_\phi}^2\dpn{\pi_\phi(f)\xi_1, \xi_1}{H_\phi} + \normn{(1 - P)\xi_\phi}_{H_\phi}^2 \dpn{\pi_\phi(f)\xi_2, \xi_2}{H_\phi} \\
+        &= \normn{P\xi_\phi}_{H_\phi}^2 \dpn{f, \psi_1}{L^1(G; \complex)} + \normn{(1 - P)\xi_\phi}_{H_\phi}^2 \dpn{f, \psi_2}{L^1(G; \complex)}
+    \end{align*}
+  
+    so $\phi$ is a strict convex combination of two states. If $\psi_1 = \phi$, then for any $f, g \in L^1(G; \complex)$, 
+    \begin{align*}
+        \dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} &= \normn{P\xi_\phi}_{H_\phi}^{-2} \dpn{P\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} \\
+        \dpn{[\pi_\phi(f) - \normn{P\xi_\phi}_{H_\phi}^{-2}P\pi_\phi(f)]\xi_\phi,\pi_\phi(g) \xi_\phi}{H_\phi} &= 0
+    \end{align*}
+
+    so $I - \normn{P\xi_\phi}_{H_\phi}^{-2}P = 0$, which contradicts the fact that $P$ is a non-trivial projection.
+\end{proof}

src/harmonic/rep/index.tex

@@ -3,4 +3,6 @@
 
 \input{./unit.tex}
 \input{./l1.tex}
+\input{./gns.tex}
+\input{./pos.tex}
 

src/harmonic/rep/pos.tex

@@ -0,0 +1,237 @@
+\section{Positive-Definite Functions}
+\label{section:positive-definite-function}
+
+
+\begin{lemma}
+\label{lemma:group-positive-computation-gymnastics}
+    Let $G$ be a locally compact group, $\phi \in L^\infty(G; \complex)$, and $f \in L^1(G; \complex)$, then
+    \[
+    \int_G (f^* * f) \phi = \iint_{G \times G} f(x)\ol{f(y)}\phi(y^{-1}x)dydx
+    \]
+\end{lemma}
+\begin{proof}
+    \begin{align*}
+        \int_G (f^* * f)(x)\phi(x) dx &= \iint_{G \times G} \Delta_G(y^{-1})\ol{f(y^{-1})}f(y^{-1}x)\phi(x)dydx \\
+        &= \iint_{G \times G} \ol{f(y)} f(yx)\phi(x)dy dx \\
+        &= \iint_{G \times G} f(x) \ol{f(y)} \phi(y^{-1}x)dydx
+    \end{align*}
+\end{proof}
+
+\begin{definition}[Positive-Definite Function]
+\label{definition:group-positive-definite-function}
+    Let $G$ be a locally compact group and $\phi \in BC(G; \complex)$, then the following are equivalent:
+    \begin{enumerate}
+        \item The mapping $f \mapsto \dpn{f, \phi}{L^1(G; \complex)}$ is a positive linear functional on $L^1(G; \complex)$. 
+        \item For each $n \in \natp$, $\seqf{\lambda_j} \subset \complex$ and $\seqf{x_j} \subset G$, 
+        \[
+        \sum_{i = 1}^n \sum_{j = 1}^n \lambda_i \ol{\lambda_j} \phi(x_j^{-1}x_i) \ge 0
+        \]
+    \end{enumerate}
+    
+    If the above holds, then $\phi$ is \textbf{positive-definite}. 
+    
+    The set $P(G)$ is the \textbf{space of positive-definite functions} on $G$. The set $P_0(G)$ is the space of positive-definite functions of norm at most 1. Finally, the set $S(G)$ of positive-definite functions of norm 1 is the \textbf{state space} of $G$. 
+\end{definition}
+\begin{proof}[Proof, {{\cite[Proposition 3.35]{FollandHarmonic}}}. ]
+    (1) $\Rightarrow$ (2): Using \autoref{proposition:group-algebra-approximate-identity}, let $\angles{\psi_\alpha}_{\alpha \in A} \subset L^1(G; \complex)$ be an approximate identity such that for all $U \in \cn_G(1)$, there exists $\alpha_0 \in A$ with $\supp{\psi_\alpha} \subset U$ for all $\alpha \ge \alpha_0$. 
+    
+    For each $\alpha \in A$, $\seqf{\lambda_j} \subset \complex$, and $\seqf{x_j} \subset G$, let $f_\alpha = \sum_{j = 1}^n \lambda_j L_{x_j}\psi_\alpha$, then by continuity of $\phi$, 
+    \begin{align*}
+        0 \le \int_G (f^*_\alpha * f_\alpha) \phi &= \sum_{i, j = 1}^n \lambda_i \ol{\lambda_j} \iint_{G \times G} \psi_\alpha(x_i^{-1}y)\ol{\psi_\alpha(x_j^{-1}z)}\phi(z^{-1}y)dydz \\
+        0 \le \lim_{\alpha \in A}\int_G (f^*_\alpha * f_\alpha) \phi &= \sum_{i, j = 1}^n \lambda_i \ol{\lambda_j}\phi(x_j^{-1}x_i)
+    \end{align*}
+    
+
+    (2) $\Rightarrow$ (1): Let $f \in C_c(G; \complex)$ and $K = \supp{f}$, then $F(x, y) = f(x)\ol{f(y)}\phi(y^{-1}x)$ is uniformly continuous by \autoref{proposition:lcg-cc-uc}. As such, there exists a partition $K = \bigsqcup_{j = 1}^n U_j$ and $\seqf{x_j} \subset G$ such that $|F(x_i, x_j) - F(x, y)| \le \eps$ for all $(x, y) \in U_i \times U_j$ and $1 \le i, j \le n$. Thus
+    \[
+    \abs{\int_G (f^* * f) \phi - \sum_{i, j = 1}^n F(x_i, x_j) \cdot |U_i| \cdot |U_j|} \le \eps |K|^2
+    \]
+
+    As
+    \[
+    \sum_{i, j = 1}^n F(x_i, x_j) \cdot |U_i| \cdot |U_j| = \sum_{i, j = 1}^n f(x_i)|U_i| \cdot \ol{f(x_j)|U_j|} \cdot \phi(x_j^{-1}x_i) \ge 0
+    \]
+
+    and the above holds for all $\eps > 0$, $\int_G (f^* * f) \phi \ge 0$. By \hyperref[density of $C_c(G; \complex)$ in $L^1(G; \complex)$]{proposition:radon-cc-dense} and continuity of the convolution product, $\int_G (g^* * g) \phi \ge 0$ for all $g \in L^1(G; \complex)$. 
+\end{proof}
+
+\begin{lemma}
+\label{lemma:positive-definite-function-conjugate}
+    Let $G$ be a locally compact group and $\phi \in BC(G; \complex)$ be a positive-definite function, then $\ol \phi$ is also positive definite. 
+\end{lemma}
+\begin{proof}[Proof, {{\cite[Proposition 3.14]{FollandHarmonic}}}. ]
+    For any $f \in L^1(G; \complex)$, 
+    \[
+    \int_G (f^* * f)\ol \phi = \int_G (\ol{f}^* * \ol{f}) \phi
+    \]
+\end{proof}
+
+\begin{lemma}
+\label{lemma:positive-definite-from-representation}
+    Let $G$ be a locally compact group, then:
+    \begin{enumerate}
+        \item For any unitary representation $(H, \pi)$ of $G$ and $\xi \in H$, $\phi(x) = \dpn{\pi(x)\xi, \xi}{H}$ is a positive-definite function.
+        \item For any $f \in L^2(G; \complex)$, let $\tilde f(x) = \ol{f(x^{-1})}$, then $f * \tilde f$ is a positive-definite function. 
+    \end{enumerate}
+\end{lemma}
+\begin{proof}
+    (1): Let $f \in L^1(G; \complex)$, then by \autoref{lemma:group-positive-computation-gymnastics}, 
+    \begin{align*}
+        \int_G (f^* * f)(x)\phi(x)dx &= \iint_{G \times G} f(x)\ol{f(y)}\phi(y^{-1}x)dxdy \\
+        &= \iint_G \dpn{f(x)\pi(x)\xi, f(y)\pi(y)\xi}{H}dxdy = \norm{\pi(f)\xi}_H^2 \ge 0
+    \end{align*}
+
+    (2): Let $\pi: G \to B(L^2(G; \complex))$ be the left regular representation of $G$, then
+    \[
+    \dpn{\pi(x)f, f}{L^2(G; \complex)} = \int_G f(x^{-1}y)\ol{f(y)}dy = \ol{(f * \tilde f)(x)}
+    \]
+
+    By (1) and \autoref{lemma:positive-definite-function-conjugate}, $f * \tilde f$ is positive-definite. 
+\end{proof}
+
+\begin{lemma}
+\label{lemma:positive-definite-cc-decompose}
+    Let $G$ be a locally compact group, then for any $f * g \in C_c(G; \complex) * C_c(G; \complex)$, there exists $\bracsn{h_k}_0^3 \subset C_c(G; \complex) \cap P(G)$ such that
+    \[
+    f * g = \sum_{k = 0}^3 i^k h_k
+    \]
+\end{lemma}
+\begin{proof}
+    For each $h \in C_c(G; \complex)$, denote $\tilde h(x) = \ol{h(x^{-1})}$, then by \hyperref[polarisation]{proposition:polarisation-complex},
+    \[
+        f * g = \frac{1}{4}\sum_{k = 0}^3 i^k (f + i^k \tilde g) * (f + i^k \tilde g)^\sim
+    \]
+    
+    Since $C_c(G; \complex) \subset L^2(G; \complex)$, (2) of \autoref{lemma:positive-definite-from-representation} implies that $(f + i^k \tilde g) * (f + i^k \tilde g)^\sim$ is positive-definite. 
+\end{proof}
+
+
+
+
+\begin{lemma}
+\label{lemma:lcg-positive-definite-function-concentration}
+    Let $G$ be a locally compact group and $\phi \in S(G)$, then for each $x, y \in G$, 
+    \[
+    |\phi(x) - \phi(y)|^2 \le 2 - 2\text{Re}\phi(yx^{-1})
+    \]
+\end{lemma}
+\begin{proof}[Proof, {{\cite[Lemma 3.30]{FollandHarmonic}}}. ]
+    Let $(H_\phi, \pi_\phi, \xi_\phi)$ be the \hyperref[GNS triple]{definition:gns-triple-lcg} associated with $\phi$, then by \autoref{corollary:group-l1-positive-function}, $\phi(x) = \dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}$ for all $x \in G$. Since $\norm{\phi}_u = \phi(1_G) = 1$, $\norm{\xi_\phi}_{H_\phi} = 1$. As such, 
+    \begin{align*}
+        |\phi(x) - \phi(y)|^2 &= |\dpn{[\pi_\phi(x) - \pi_\phi(y)]\xi_\phi, \xi_\phi}{H_\phi}|^2 \\
+        &=  |\dpn{\xi_\phi, [\pi_\phi(x^{-1}) - \pi_\phi(y^{-1})]\xi_\phi}{H_\phi}|^2 \\
+        &\le \normn{[\pi_\phi(x^{-1}) - \pi_\phi(y^{-1})]\xi_\phi}_{H_\phi}^2 \\
+        &= 2 - 2\text{Re}\dpn{\pi_\phi(x^{-1})\xi_\phi, \pi_\phi(y^{-1})\xi_\phi}{H_\phi} \\
+        &= 2 - 2\text{Re}\dpn{\pi_\phi(yx^{-1})\xi_\phi, \xi_\phi}{H_\phi} = 2 - 2\text{Re}\phi(yx^{-1})
+    \end{align*}
+\end{proof}
+
+
+\begin{theorem}
+\label{theorem:group-bc-weak-topology}
+    Let $G$ be a locally compact group, then the topology of uniform convergence on compact subsets of $G$ on $S(G)$ and the weak* topology on $S(G)$ as a subset of $L^1(G; \complex)^*$ coincide. 
+\end{theorem}
+\begin{proof}
+    ($\supset$): Let $\seqf{f_j} \subset L^1(G; \complex)$. By inner regularity of the Haar measure and hence of $|f_j(x)|dx$, there exists $K \subset G$ compact such that $\int_{G \setminus K}|f_j| < \eps$ for all $1 \le j \le n$. In which case, for any $\phi, \psi \in S(G)$, 
+    \begin{align*}
+        \max_{1 \le j \le n}\abs{\int_{G} f_j (\phi - \psi)} &\le \max_{1 \le j \le n}\int_G|f_j(\phi - \psi)| \\
+        &\le \max_{1 \le j \le n}\braks{\int_{K} |f_j(\phi - \psi)| + \int_{G \setminus K}|f_j(\phi - \psi)|} \\
+        &\le \sup_{x \in K}|\phi(x) - \psi(x)|\max_{1 \le j \le n}\norm{f_j}_{L^1(G; \complex)} + 2\eps
+    \end{align*}
+
+    ($\subset$): Let $\phi \in S(G)$ and $V \in \cn_G(1_G)$ be relatively compact, then for each $x \in G$, 
+    \begin{align*}
+        \abs{\phi(x) - \frac{\one_V}{|V|} * \phi(x)} &\le \frac{1}{|V|}\int_V |\phi(x) - \phi(y^{-1}x)|dy \\
+        &\le \frac{1}{|V|}\int_V ({2 - 2\text{Re}(\phi(y))})^{1/2} dy \\
+        &\le |V|^{-1/2}\braks{\int_V 2 - 2\text{Re}(\phi(y)) dy}^{1/2}
+    \end{align*}
+
+    by \autoref{lemma:lcg-positive-definite-function-concentration} and the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}. As such, for any $\phi, \psi \in S(G)$ and $x \in G$,
+    \begin{align*}
+        |\phi(x) - \psi(x)| &\le \abs{\phi(x) - \frac{\one_V}{|V|} * \phi(x)} + \frac{1}{|V|}\abs{\one_V * (\phi - \psi)(x)}\\
+        &+ \abs{\psi(x) - \frac{\one_V}{|V|} * \psi(x)} \\
+        &\le |V|^{-1/2}\braks{\int_V 2 - 2\text{Re}(\phi(y)) dy}^{1/2} + \abs{\angles{\frac{L_x\one_V}{|V|}, \phi - \psi}_{L^1(G; \complex)}} \\
+        &+ |V|^{-1/2}\braks{\int_V 2 - 2\text{Re}(\psi(y)) dy}^{1/2} \\
+        &\le 2|V|^{-1/2}\braks{\int_V 2 - 2\text{Re}(\phi(y)) dy}^{1/2}+ 2|V|^{-1/2}|\dpn{\one_V, \phi - \psi}{L^1(G; \complex)}|^{1/2} \\
+        &+ \abs{\angles{\frac{L_x\one_V}{|V|}, \phi - \psi}_{L^1(G; \complex)}} \\
+        &\le 4\sup_{y \in V}|\phi(y) - 1|^{1/2}+ |2\dpn{\one_V, \phi - \psi}{L^1(G; \complex)}|^{1/2}\\
+        &+ \abs{\angles{\frac{L_x\one_V}{|V|}, \phi - \psi}_{L^1(G; \complex)}} \\
+    \end{align*}
+
+    For any $K \subset G$ compact,
+    \begin{align*}
+    \sup_{x \in K}|\phi(x) - \psi(x)| &\le 4\sup_{y \in V}|\phi(y) - 1|^{1/2}+ |2\dpn{\one_V, \phi - \psi}{L^1(G; \complex)}|^{1/2} \\
+    &+\sup_{x \in K}\abs{\angles{\frac{L_x\one_V}{|V|}, \phi - \psi}_{L^1(G; \complex)}}
+    \end{align*}
+
+    Let $\eps > 0$. By \autoref{corollary:group-l1-positive-function}, $\phi$ is continuous with $\phi(1_G) = 1$, so there exists a relatively compact neighbourhood $V \in \cn_G(1_G)$ with $4\sup_{y \in V}|\phi(y) - 1|^{1/2} < \eps$. 
+    
+    Since $S(G)$ is a bounded subset of $L^1(G; \complex)^*$, it is equicontinuous. By the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, the weak* topology and the topology of uniform convergence on compact subsets of $G$ coincide on $S(G)$. In particular, given that $K$ is compact, $\bracsn{L_x\one_V|x \in K}$ is compact in $L^1(G; \complex)$, so there exists $W \in \cn_{S(G)}(\phi)$ such that
+    \[
+    |2\dpn{\one_V, \phi - \psi}{L^1(G; \complex)}|^{1/2} +\sup_{x \in K}\abs{\angles{\frac{L_x\one_V}{|V|}, \phi - \psi}_{L^1(G; \complex)}} \le \eps
+    \]
+
+    for all $\psi \in W$. Therefore for any $\psi \in W$, $\sup_{x \in K}|\phi(x) - \psi(x)|  \le 2\eps$. 
+
+    
+\end{proof}
+
+
+\begin{theorem}
+\label{theorem:lcg-state-extreme}
+    Let $G$ be a locally compact group, then:
+    \begin{enumerate}
+        \item The extreme points of $P_0(G)$ are the extreme points of $S(G)$ and $0$. 
+        \item $S(G)$ is contained in the closed convex hull of its extreme points\footnote{In general, $L^1(G; \complex)$ is not unital and $S(G)$ is not closed.}. 
+    \end{enumerate}
+\end{theorem}
+\begin{proof}[Proof, {{\cite[Lemma 3.26, Theorem 3.27]{FollandHarmonic}}}. ]
+    (1): Let $\phi \in S(G)$, $\psi_1, \psi_2 \in P_0(G)$, and $t \in (0, 1)$ such that $\phi = (1 - t)\psi_1 + t\psi_2$, then
+    \[
+    1 = \norm{\phi}_{L^1(G; \complex)^*} \le (1 - t)\norm{\psi_1}_{L^1(G; \complex)^*} + t\norm{\psi_2}_{L^1(G; \complex)^*} \le 1
+    \]
+
+    so $\norm{\psi_1}_{L^1(G; \complex)^*} = \norm{\psi_2}_{L^1(G; \complex)^*} = 1$, and $S(G)$ is an extreme subset of $P_0(G)$. By \autoref{lemma:extreme-subset-extreme-point}, every extreme point of $S(G)$ is an extreme point of $P_0(G)$. 
+
+    On the other hand, every extreme point of $P_0(G)$ in $S(G)$ is an extreme point of $S(G)$. 
+    
+    It remains to show that $0$ is the only other extreme point of $P_0(G)$. To this end, let $\phi, \psi \in P_0(G)$ and $t \in (0, 1)$ such that $0 = (1 - t)\phi + t\psi$, then $0 = (1 - t)\phi(1_G) + t\psi(1_G)$. By \autoref{corollary:group-l1-positive-function}, $0 = \phi(1_G) = \norm{\phi}_u$ and $0 = \psi(1_G) = \norm{\psi}_u$ so $0$ is an extreme point of $P_0(G)$. For any $\phi \in P_0(G) \setminus (S(G) \cup \bracsn{0})$, $\phi/\phi(1_G) \in S(G)$, and $\phi = (1 - \phi(1_G)) \cdot 0 + \phi(1_G) \cdot \phi/\phi(1_G)$. Therefore $0$ is the only other extreme point of $P_0(G)$. 
+
+    (2): Let $\phi \in S(G)$ and $\eps \in (0, 1)$. By definition of the operator norm and (4) of \autoref{proposition:semicontinuous-properties}, $\norm{\cdot}_{L^1(G; \complex)^*}$ is lower semicontinuous on $L^1(G; \complex)^*$ with respect to the weak* topology, so $\bracsn{\psi \in P_0(G)|\ \norm{\psi}_{L^1(G; \complex)^*} > 1 - \eps}$ is a weak*-neighbourhood of $\phi$. 
+
+    Let $\cf \subset L^1(G; \complex)$ be finite, then by (1) and the \hyperref[Krein-Milman Theorem]{theorem:krein-milman}, there exist extreme points $\seqf{\phi_j} \subset S(G)$ and $\bracsn{\lambda_j}_0^n \subset [0, 1]$ such that $\sum_{j = 0}^n \lambda_j = 1$, $\lambda_0 < \eps/\max_{f \in \cf}\norm{f}_{L^1(G; \complex)}$, and
+    \[
+    \max_{f \in \cf}\angles{f, \phi - \sum_{j = 1}^n \lambda_j \phi_j}_{L^1(G; \complex)} < \eps
+    \]
+
+    In which case, $\frac{1}{1 - \lambda_0}\sum_{j = 1}^n \lambda_j = 1$ and
+    \[
+    \max_{f \in \cf}\angles{f, \sum_{j = 1}^n \lambda_j \phi_j - \frac{1}{1 - \lambda_0}\sum_{j = 1}^n \lambda_j \phi_j}_{L^1(G; \complex)} \le |\lambda_0| \max_{f \in \cf}\norm{f}_{L^1(G; \complex)} \le \eps
+    \]
+    
+    so
+    \[
+    \max_{f \in \cf}\angles{f, \phi - \sum_{j = 1}^n \frac{\lambda_j}{1 - \lambda_0} \phi_j}_{L^1(G; \complex)} < 2\eps
+    \]
+\end{proof}
+
+
+\begin{theorem}[Gelfand-Raikov]
+\label{theorem:gelfand-raikov-lcg}
+    Let $G$ be a locally compact group and $x, y \in G$ be distinct, then there exists an irreducible unitary representation $(H, \pi)$ of $G$ such that $\pi(x) \ne \pi(y)$. 
+\end{theorem}
+\begin{proof}[Proof, {{\cite[Theorem 3.34]{FollandHarmonic}}}. ]
+    Let $x, y \in G$ and suppose that for any irreducible unitary representation $(H, \pi)$ of $G$, $\pi(x) = \pi(y)$. Let $\phi \in S(G)$ be an extreme point of $S(G)$ and $(H_\phi, \pi_\phi, \xi_\phi)$ be its \hyperref[GNS triple]{definition:gns-triple-lcg}, then $(H_\phi, \pi_\phi)$ is irreducible by \autoref{proposition:gns-irreducible-lcg}. In particular, by \autoref{corollary:group-l1-positive-function}, 
+    \[
+    \phi(x) = \dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi} = 
+    \dpn{\pi_\phi(y)\xi_\phi, \xi_\phi}{H_\phi} = \phi(y)
+    \]
+
+    By (2) of \autoref{theorem:lcg-state-extreme}, the convex hull of extreme points of $S(G)$ is weak*-dense in $S(G)$. By \autoref{theorem:group-bc-weak-topology}, the weak* topology and the topology of uniform convergence on compact subsets of $G$ coincide on $S(G)$. As such, $\psi(x) = \psi(y)$ for all $\psi \in S(G)$. By linearity, $\psi(x) = \psi(y)$ for all $\psi \in P(G)$ as well. 
+
+    By linearity and \autoref{lemma:positive-definite-cc-decompose}, $(f * g)(x) = (f * g)(y)$ for all $f, g \in C_c(G; \complex)$. Using \autoref{proposition:group-algebra-approximate-identity}, let $\angles{\psi_U}_{U \in \cn_G(1_G)} \subset C_c(G; \complex)$ be an approximate identity for $L^1(G; \complex)$ such that $\supp{\psi_U} \subset U$ for all $U \in \cn_G(1_G)$, then for any $g \in C_c(G; \complex)$, $\psi_U * g \to g$ uniformly. Therefore $g(x) = g(y)$ for all $g \in C_c(G; \complex)$. 
+    
+    Should $x \ne y$, then \hyperref[Urysohn's Lemma]{lemma:lch-urysohn} implies that there exists $g \in C_c(G; \complex)$ with $g(x) \ne g(y)$. Since this is not the case, $x = y$. 
+\end{proof}
+
+

src/measure/measure/complete.tex

@@ -17,7 +17,7 @@
     \begin{enumerate}
         \item For any $A \in \ol{\cm}$, there exists $E \in \cm$, $N \in \cn$, and $F \subset N$ such that $E \cap N = \emptyset$ and $A = E \sqcup F$.
         \item $\ol{\cm} \supset \cm$ is a $\sigma$-algebra.
-        \item There exists an extension $\ol{\mu}$ of $\mu$ as a measure on $\cm$.
+        \item There exists an extension $\ol{\mu}$ of $\mu$ as a measure on $\ol \cm$.
         \item $(X, \ol{\cm}, \ol{\mu})$ is a complete measure space.
         \item[(U)] For any complete measure space $(X, \cf, \nu)$ where $\cf \supset \cm$ and $\nu|_\cm = \mu$, $\cf \supset \ol{\cm}$ and $\nu|_{\ol{\cm}} = \ol{\mu}$.
     \end{enumerate}
@@ -30,9 +30,9 @@
     (2): Let $A \in \ol{\cm}$, $E \in \cm$, $N \in \cn$, and $F \subset N$ such that $A = E \sqcup F$, then
     \begin{align*}
         A^c &= (A^c \cap N^c) \cup (A^c \cap N) = (N^c \setminus A) \cup (N \setminus A) \\
-        &= \underbrace{(N^c \setminus E)}_{\in \cm} \cup \underbrace{(N \setminus F)}_{\subset N} \in \cm
+        &= \underbrace{(N^c \setminus E)}_{\in \cm} \cup \underbrace{(N \setminus F)}_{\subset N} \in \ol \cm
     \end{align*}
-    Let $\seq{A_n} \subset \cm$. For each $n \in \natp$, let $E_n \in \cm$, $N_n \in \cn$, and $F_n \subset N$ such that $A_n = E_n \cup F_n$, then
+    Let $\seq{A_n} \subset \cm$. For each $n \in \natp$, let $E_n \in \cm$, $N_n \in \cn$, and $F_n \subset N_n$ such that $A_n = E_n \cup F_n$, then
     \[
     \bigcup_{n \in \natp}A_n = \underbrace{\braks{\bigcup_{n \in \natp}E_n}}_{\in \cm} \cup \underbrace{\braks{\bigcup_{n \in \natp}F_n}}_{\subset \bigcup_{n \in \natp}N_n} \in \ol{\cm}
     \]

src/measure/sets/elementary.tex

@@ -7,7 +7,7 @@
     \begin{enumerate}
         \item[(P1)] $\emptyset \in \ce$.
         \item[(P2)] For any $A, B \in \ce$, $A \cap B \in \ce$.
-        \item[(E)] For any $E, F \in \ce$ with $E \subset F$, there exists $\seqf{E_j} \subset \ce$ such that $E \setminus F = \bigsqcup_{j = 1}^n E_j$.
+        \item[(E)] For any $E, F \in \ce$ with $E \subset F$, there exists $\seqf{E_j} \subset \ce$ such that $F \setminus E = \bigsqcup_{j = 1}^n E_j$.
     \end{enumerate}
 
     If $X \in \ce$, then (E) may be replaced with

src/op/c-star/gns.tex

@@ -61,9 +61,9 @@
 
     For each $x \in A$, let $\pi_\phi(x)$ be the continuous extension of $\pi_\phi^0(x)$ to an element of $B(H_\phi)$, then:
     \begin{enumerate}
-        \item $(H_\phi, \dpn{\cdot, \cdot}{\phi})$ is a Hibert space.
+        \item $H_\phi$ equipped with the unique continuous extension of $\dpn{\cdot, \cdot}{\phi}$ is a Hibert space.
         \item $(H_\phi, \pi_\phi)$ is a well-defined representation of $A$. 
-        \item $\xi_\phi = 1_A + N_\phi$ is a unit vector in $H_\phi$, and $\bracsn{\pi_\phi(x)\xi_\phi| x \in A}$ is dense in $H_\phi$. Moreover, for each $x, y \in A$,
+        \item $\xi_\phi = 1_A + N_\phi$ is a unit cyclic vector in $H_\phi$ such that for each $x, y \in A$,
         \[
         \dpn{x, y}{\phi} = \dpn{\pi_\phi(x)\xi_\phi, \pi_\phi(y)\xi_\phi}{H_\phi}
         \]
@@ -108,7 +108,7 @@
 
 \begin{proposition}
 \label{proposition:gns-irreducible}
-    Let $A$ be a unital $C^*$ algebra, $\phi \in S(A)$ be a state, and $(H, \pi, \xi)$ be its GNS triple, then $\phi$ is a pure state if and only if $(H, \pi)$ is irreducible. 
+    Let $A$ be a unital $C^*$-algebra, $\phi \in S(A)$ be a state, and $(H, \pi, \xi)$ be its GNS triple, then $\phi$ is a pure state if and only if $(H, \pi)$ is irreducible. 
 \end{proposition}
 \begin{proof}
     ($\Rightarrow$): Suppose that $(H, \pi)$ is reducible, then there exists a non-trivial invariant subspace $M \subset H$. Let $P \in B(H)$ be the orthogonal projection onto $M$, $\xi_1 = P\xi/\norm{P\xi}_H$, and $\xi_2 = (\xi - P\xi)/\norm{\xi - P\xi}_H$. For each $x \in A$, let
@@ -118,14 +118,21 @@
 
     then $\phi_1$ and $\phi_2$ are vector states on $A$. Since $M$ and $M^\perp$ are invariant for $\pi$, 
     \begin{align*}
-        \dpn{x, \phi}{A} &= \dpn{\pi(x)\xi, \xi}{A} = \dpn{\pi(x)P\xi, P\xi}{A} + \dpn{\pi(x)(I - P)\xi, (I - P)\xi}{A} \\
+        \dpn{x, \phi}{A} &= \dpn{\pi(x)\xi, \xi}{H} = \dpn{\pi(x)P\xi, P\xi}{H} + \dpn{\pi(x)(I - P)\xi, (I - P)\xi}{H} \\
-        &= \norm{P\xi}_H^2\dpn{\pi(x)\xi_1, \xi_1}{H} + \norm{(1 - P)\xi}_H^2 \dpn{\pi(x)\xi_2, \xi_2}{H} \\
+        &= \norm{P\xi}_H^2\dpn{\pi(x)\xi_1, \xi_1}{H} + \norm{(I - P)\xi}_H^2 \dpn{\pi(x)\xi_2, \xi_2}{H} \\
-        &= \norm{P\xi}_H^2 \dpn{x, \phi_1}{A} + \norm{(1 - P)\xi}_H^2 \dpn{x, \phi_2}{A}
+        &= \norm{P\xi}_H^2 \dpn{x, \phi_1}{A} + \norm{(I - P)\xi}_H^2 \dpn{x, \phi_2}{A}
     \end{align*}
 
-    so $\phi$ is a strict convex combination of two states, and is not pure.
+    so $\phi$ is a strict convex combination of two states. If $\phi_1 = \phi$, then for any $x, y \in A$, 
+    \begin{align*}
+    \dpn{\pi(x)\xi, \pi(y)\xi}{H} &= \norm{P\xi}_H^{-2}\dpn{\pi(x)P\xi, \pi(y)\xi}{H} \\
+    \dpn{[\pi(x) - \norm{P\xi}_H^{-2}P\pi(x)]\xi, \pi(y)\xi}{H} &= 0
+    \end{align*}
 
-    ($\Leftarrow$): Suppose that $(H, \pi)$ is irreducible. Let $t \in (0, 1)$ and $\phi_1, \phi_2 \in S(A)$ with $\phi = t\phi_1 + (1 - t)\phi_2$. Let $T: \pi(A)H \to H$ be defined by
+    so $I - \norm{P\xi}_H^{-2}P = 0$, which contradicts the fact that $P$ is a non-trivial projection. 
+    
+
+    ($\Leftarrow$): Suppose that $(H, \pi)$ is irreducible. Let $t \in (0, 1)$ and $\phi_1, \phi_2 \in S(A)$ with $\phi = t\phi_1 + (1 - t)\phi_2$. Let $T: \pi(A)\xi \to H$ be defined by
     \[
     \dpn{T\pi(x)\xi, \pi(y)\xi}{H} = \dpn{x, y}{\phi_1} = \dpn{y^*x, \phi_1}{A}
     \]
@@ -133,7 +140,7 @@
     then since $\phi_1 \le t^{-1}\phi$, $T$ extends continuously into a positive operator in $B(H)$. Moreover, for any $x, y, z \in A$, 
     \begin{align*}
     \dpn{\pi(z)T\pi(x)\xi, \pi(y)\xi}{H} &= \dpn{T\pi(x)\xi, \pi(z^*y)\xi}{H} \\
-    &= \dpn{y^*zx, \phi_1}{A} = \dpn{T\pi(z)\pi(x)\xi, \xi}{H}
+    &= \dpn{y^*zx, \phi_1}{A} = \dpn{T\pi(z)\pi(x)\xi, \pi(y)\xi}{H}
     \end{align*}
     
     so $T \in \pi(A)'$. By \hyperref[Schur's Lemma]{lemma:cstar-schur}, there exists $\lambda \in \complex$ such that $T = \lambda I$. However, since $\dpn{T\xi, \xi}{H} = \dpn{1_A, \phi_1}{A} = 1$, $\lambda = 1$, $T = I$, and $\phi_1 = \phi$. By symmetry, $\phi_2 = \phi$ as well. 

src/topology/main/compactify.tex

@@ -27,7 +27,7 @@
 
     Moreover, if $(\beta X, e)$ is \textit{any} pair that satisfies (U1), then
     \begin{enumerate}
-        \item[(U2)] For any pair $(Y, \varphi)$ satisfying (1), there exists a unique $\beta \varphi \in C(\beta X; Y)$ such that the following diagram commutes:
+        \item[(U2)] For any compact Hausdorff space $Y$ and $\varphi \in C(X; Y)$, there exists a unique $\beta \varphi \in C(\beta X; Y)$ such that the following diagram commutes:
         \[
         \xymatrix{
         \beta X \ar@{->}[r]^{\beta \varphi} & Y \\
@@ -89,6 +89,7 @@
         X \ar@{->}[u]^{e} \ar@{->}[ru]_{f} & 
         }
         \]
+        \item[(F)] For any discrete space $Y$, $f: X \to Y$, and $\fU \in \beta X$, $\beta f(\fU)$ is the ultrafilter generated by $f(\fU)$. 
     \end{enumerate}
 
     In other words, $(\beta X, e)$ as constructed above is the Stone-Čech compactification of $X$. 
@@ -103,10 +104,10 @@
     \bigcap_{A \in F}U_{A^c} = \bigcap_{A \in F}U_A^c = \braks{\bigcup_{A \in F}U_A}^c \ne \emptyset
     \]
     
-    In particular, $\bigcap_{A \in F}A^c \ne \emptyset$. As such, $\bracsn{A^c|A \in \mathcal{S}}$ generates an ultrafilter $\fU \in \beta X$. Since $\fU \not\in U_A$ for all $A \in \mathcal{S}$, $\mathcal{U}$ does not cover $X$, which is a contradiction. Thus $\beta X$ is compact. 
+    In particular, $\bigcap_{A \in F}A^c \ne \emptyset$. By the \hyperref[ultrafilter lemma]{lemma:ultrafilter}, $\bracsn{A^c|A \in \mathcal{S}}$ generates an ultrafilter $\fU \in \beta X$. Since $\fU \not\in U_A$ for all $A \in \mathcal{S}$, $\mathcal{U}$ does not cover $X$, which is a contradiction. Thus $\beta X$ is compact. 
 
     Finally, since $X$ is discrete, $e$ is automatically continuous. On the other hand, for each $x \in X$, $U_{\bracsn{x}} = e(\bracsn{x})$ is open in $\beta X$. In addition, for any $\emptyset \ne A \subset X$, $e(x) \in U_A$ for all $x \in A$, so $U_A \cap e(X) \ne \emptyset$, and $e(X)$ is dense in $\beta X$. Therefore $e(X)$ is embedded as a dense subspace of $\beta X$. 
-
+  
     (U1): Let $f \in C(X; [0, 1])$, and let $\beta f: \beta X \to [0, 1]$ be defined by $\beta f(\fU) = \lim_{\fU}f$. Fix $\fU \in \beta X$ and $\eps > 0$, then there exists $A \in \fU$ such that $f(A) \subset B_{[0, 1]}(\lim_{\fU}f, \eps)$. Thus for any $\fV \in U_A$, 
     \[
     \beta f(\fV) = \lim_{\fV}f \in \ol{f(A)} \subset B_{[0, 1]}\paren{\lim_{\fU}f, \eps}

src/topology/main/stonean.tex

@@ -52,4 +52,29 @@
     Given that $\one_U \le F \le \one_{\ol{U}}$ and $F$ is continuous, $\bracs{F = 1} \supset \ol{U}$, so $F = \one_{\ol{U}}$ is continuous. Hence $\ol{U} = F^{-1}(1)$ must be open. 
 \end{proof}
 
+\begin{theorem}[Gleason, 1958]
+\label{theorem:gleason-extremally-disconnected}
+    Let $X$ be a compact Hausdorff space, then the following are equivalent:
+    \begin{enumerate}
+        \item $X$ is extremally disconnected. 
+        \item[(P)] For any compact Hausdorff spaces $Y, Z$, surjective continuous map $\pi \in C(Y; Z)$, and $f \in C(X; Z)$, there exists a lifting $\tilde f \in C(X; Y)$ such that the following diagram commutes: 
+        \[
+        \xymatrix{
+        & Y \ar@{->}[d]^{\pi} \\
+        X \ar@{->}[ru]^{\tilde f} \ar@{->}[r]^{f} & Z
+        }
+        \]
+        \item[(I)] For every pair of operator systems $S \subset T$ and completely positive map $\phi: S \to C(X; \complex)$, there exists a completely positive map $\Phi: T \to C(X; \complex)$ such that the following diagram commutes: 
+        \[
+        \xymatrix{
+        T \ar@{->}[r]^{\Phi} & C(X; \complex) \\
+        S \ar@{^{(}->}[u] \ar@{->}[ru]_{\phi} & 
+        }
+        \]
+    \end{enumerate}
+\end{theorem}
+\begin{proof}
+    See \cite{GleasonProjective}. 
+\end{proof}
+