Added a proof for Kadison-Schwarz.

src/dyn/top/def.tex

@@ -72,7 +72,7 @@
 
 \begin{definition}[Pointed Flow]
 \label{definition:tds-pointed}
-    Let $G$ be a group, then a \textbf{pointed $G$-flow} is a pair $(X, x_0)$, where $X$ is a $G$-flow, and $x_0 \in X$ is a distinguished point.
+    Let $G$ be a group, then a \textbf{pointed $G$-flow} is a pair $(X, x_0)$, where $X$ is a $G$-flow, and $x_0 \in X$ is a distinguished point. The mapping $\rho_x: G \to X$ defined by $g \mapsto gx_0$ is the \textbf{orbit map} of $(X, x_0)$. 
 
     Let $(G \curvearrowleft X, x_0)$ and $(G \curvearrowright Y, y_0)$ be pointed $G$-flows, then a \textbf{morphism of pointed $G$-flows} is a morphism of $G$-flows $\pi: X \to Y$ such that $\pi(x_0) = y_0$. 
 \end{definition}

src/dyn/top/recurrence.tex

@@ -14,6 +14,7 @@
     \]
 \end{definition}
 
+
 \begin{definition}[Recurrence]
 \label{definition:tds-recurrence}
     Let $G$ be an infinite group, $G \curvearrowright X$ be a $G$-flow, and $x \in X$, then $x$ is \textbf{recurrent} if for every $U \in \cn_X(x)$, $N(x, U)$ is infinite.

src/dyn/top/stone-cech.tex

@@ -1,4 +1,4 @@
-\section{$G \curvearrowright \beta G$}
+\section{The Universal Flow}
 \label{section:stone-cech-flow}
 
 \begin{definition}[Greatest Ambit]
@@ -52,3 +52,97 @@
     for all $g \in G$. Finally, since $\beta G$ is compact and $X$ is Hausdorff, $\pi(\beta G)$ is \hyperref[also compact]{proposition:compact-extensions} and \hyperref[hence closed]{proposition:compact-closed}. As $(X, x_0)$ is an ambit and $\pi(\beta G) \supset Gx_0$, $\pi(\beta G) \supset \ol{Gx_0} = X$. 
 \end{proof}
 
+\begin{definition}
+\label{definition:stone-cech-semigroup}
+    Let $G$ be a discrete group. For each $\fU, \fV \in \beta G$, let $\fU \cdot \fV = \lim_{g \to \fU}g\fV$, then
+    \begin{enumerate}
+        \item For each $A \subset G$, $A \in \fU \cdot \fV$ if and only if $\bracsn{g \in G|g^{-1}A \in \fV} \in \fU$. 
+        \item $\beta G$ equipped with the composition law $(\fU, \fV) \mapsto \fU \cdot \fV$ is a semigroup with identity $1_G$. 
+        \item (\textbf{Right Topological}) For each $\fV \in \beta G$, the mapping $\beta G \to \beta G$ defined by $\fU \mapsto \fU \cdot \fV$ is continuous. 
+    \end{enumerate}
+\end{definition}
+\begin{proof}
+    (3): By (U2) of the \hyperref[Stone-Čech compactification]{definition:stone-cech}. 
+\end{proof}
+
+\begin{definition}[Induced Action]
+\label{definition:stone-cech-semigroup-induced-action}
+    Let $G$ be a group, $(X, x_0)$ be a pointed $G$-flow, and $p_x: G \to X$ be the orbit map, then the mapping
+    \[
+    \pi: \beta G \to C(X; X) \quad \pi_\fU x  = \fU x = \lim_{g \to \fU}gx
+    \]
+
+    is a semigroup action of $\beta G$ on $X$, known as the \textbf{induced action} of $\beta G$ on $X$. 
+\end{definition}
+
+
+
+\begin{proposition}
+\label{proposition:tds-return-set-stone-cech}
+    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$. For each $y \in X$ and $\fU \in \beta G$, then $y = \fU x$ if and only if $N(x, U) \in \fU$.
+\end{proposition}
+\begin{proof}
+    By definition of filter limits, $y = \fU x$ if and only if $p_x^{-1}(U) \in \fU$ for all $U \in \cn_X(y)$. Since $p_x^{-1}(U) = N(x, U)$, $y = \fU x$ if and only if $N(x, U) \in \fU$. 
+\end{proof}
+
+\begin{proposition}
+\label{proposition:z-stone-cech-fixed-point}
+    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then the following are equivalent:
+    \begin{enumerate}
+        \item For each $U \in \cn_X(x)$, $N(x, U)$ is infinite. 
+        \item There exists $\fU \in \beta G \setminus G$ such that $px = x$. 
+    \end{enumerate}
+\end{proposition}
+% "obvious"
+
+
+\begin{definition}[Left Ideal]
+\label{definition:left-ideal-semigroup}
+    Let $G$ be a semigroup and $\emptyset \ne I \subset G$, then $I$ is a \textbf{left ideal} if for any $g \in G$ and $h \in I$, $gh \in I$. The left ideal $I$ is \textbf{minimal} if there exists no left ideal $J \subsetneq I$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:stone-cech-ideal-subflow}
+    Let $G$ be a group and $\emptyset \ne I \subset \beta G$. If $I$ is closed, then the following are equivalent:
+    \begin{enumerate}
+        \item $L$ is a $G$-subflow of $\beta G$. 
+        \item $L$ is a closed left ideal of $\beta G$. 
+    \end{enumerate}
+
+    Moreover, for arbitrary $\emptyset \ne I \subset \beta G$, the following are equivalent
+    \begin{enumerate}[label=(M\arabic*)]
+        \item $L$ is a minimal $G$-subflow of $\beta G$.
+        \item $L$ is a minimal left ideal of $\beta G$. 
+    \end{enumerate}
+    
+    In particular, minimal left ideals of $\beta G$ are closed. 
+\end{proposition}
+
+\begin{corollary}
+\label{corollary:stone-cech-minimal-ideal-existence}
+    Let $G$ be a group, then $\beta G$ admits a minimal ideal. 
+\end{corollary}
+\begin{proof}
+    By \autoref{theorem:tds-minimal-existence} and \autoref{proposition:stone-cech-ideal-subflow}. 
+\end{proof}
+
+\begin{definition}[Universal Minimal Flow]
+\label{definition:universal-minimal-flow}
+    Let $G$ be a group, then there exists a flow $(M, \alpha)$ such that:
+    \begin{enumerate}
+        \item $(M, \alpha)$ is a minimal $G$-flow. 
+        \item[(U)] For any minimal $G$-flow $(X, \beta)$, $x_0 \in X$, and $m_0 \in M$, there exists a unique continuous map $\Phi \in C(M; X)$ such that $\Phi(m_0) = x_0$ and the following diagram commutes
+        \[
+        \xymatrix{
+        M \ar@{->}[r]^{\Phi} \ar@{->}[d]_{\alpha(g)} & X \ar@{->}[d]^{\beta(g)} \\
+        M \ar@{->}[r]_{\Phi} & X
+        }
+        \]
+
+        for all $g \in G$.
+    \end{enumerate}
+\end{definition}
+
+
+
+

src/op/system/completely.tex

@@ -259,7 +259,7 @@
     \phi(x)^*\phi(x) \le \norm{\phi(1_A)}_B \phi(x^*x)
     \]
 \end{proposition}
-\begin{proof}
+\begin{proof}[Proof, general case. ]
     Since
     \[
     \begin{bmatrix}
@@ -290,6 +290,14 @@
 
     By \autoref{lemma:norm-matrix-test}, $\phi(x)^*\phi(x) \le \norm{\phi(1_A)}_B\phi(x^*x)$. 
 \end{proof}
+\begin{proof}[Proof, completely positive case. ]
+    Using the \hyperref[Gelfand-Naimark Theorem]{theorem:gns}, let $H$ be a complex Hilbert space such that $B \subset B(H)$. Assume in addition that $\phi$ is completely positive. Let $(K, V, \pi)$ be a \hyperref[Stinespring representation]{theorem:stinespring-dilation} for $\phi$, then for each $a \in A$ and $\xi \in H$, 
+    \begin{align*}
+        \dpn{\phi(a^*)\phi(a)\xi, \xi}{H} &= \dpn{V^*\pi(a^*)VV^*\pi(a)V\xi, \xi}{H} = \dpn{\pi(a^*)VV^*\pi(a)V\xi, V\xi}{K} \\
+        &\le \norm{VV^*}_{B(K)}\dpn{\pi(a^*)\pi(a)V\xi, V\xi}{K} \\
+        &\le \norm{V}_{B(K)}^2 \dpn{\pi(a^*a)V\xi, V\xi}{K} = \norm{\phi(1_A)}_{B(H)}\dpn{\phi(a^*a)\xi, \xi}{K}
+    \end{align*}
+\end{proof}
 
 
 

src/op/system/dilation.tex

@@ -49,86 +49,108 @@
 
 \begin{theorem}[Stinespring's Dilation Theorem]
 \label{theorem:stinespring-dilation}
-    Let $A$ be a unital $C^*$-algebra, $H$ be a complex Hilbert space, $\phi: A \to B(H)$ be a completely positive map, then there exists a complex Hilbert space $K$, a unital *-homomorphism $\pi: A \to B(K)$, and $V \in L(H; K)$ such that
+    Let $A$ be a unital $C^*$-algebra, $H$ be a complex Hilbert space, and $\phi: A \to B(H)$ be a completely positive map, then there exists a triple $(K, V, \pi)$, where:
+    \begin{enumerate}[label=(\alph*)]
+        \item $K$ is a complex Hilbert space. 
+        \item $V: H \to K$ is a continuous linear map.
+        \item $\pi: A \to B(K)$ is a unital *-homomorphism.
+    \end{enumerate}
+
+    such that:
     \begin{enumerate}
-        \item $\normn{\phi(1_A)}_{B(H)} = \norm{V}_{L(H; K)}^2$. 
+        \item $\norm{\phi(1_A)}_{B(H)} = \norm{V}_{L(H; K)}^2$. 
-        \item For all $a \in A$, $\phi(a) = V^*\pi(a)V$. 
+        \item For all $a \in A$, $\phi(a) = V^*\phi(a)V$. 
     \end{enumerate}
 
-    Moreover, if $\phi$ is unital, then $V$ is an isometry, and
+    Moreover, if $\phi$ is unital, then:
     \begin{enumerate}[start=2]
-        \item Under the identification of $H = V(H) \subset K$, $(K, \pi(a))$ is a dilation of $\phi(a)$ for all $a \in A$. 
+        \item $V$ is an isometry. 
+        \item Under the identification that $H = V(H) \subset K$, $(K, \pi(a))$ is a dilation of $\phi(a)$ for all $a \in A$. 
     \end{enumerate}
     
-    If $\phi$ is a state, then $(K, \pi)$ is the GNS representation of $A$ associated with $\phi$. 
+    The triple $(K, V, \pi)$ is a \textbf{Stinespring representation} of $\phi$. If $\bracsn{\pi(a)VH|a \in A}$ is dense in $K$, then $(K, V, \pi)$ is \textbf{minimal}.
-
-    The triple $(K, V, \pi)$ is a \textbf{Stinespring representation} for $\phi$. If $\bracsn{\pi(a)VH|a \in A}$ is dense in $K$, then $(K, V, \pi)$ is \textbf{minimal}.  
 \end{theorem}
 \begin{proof}[Proof, {{\cite[Theorem 4.1]{Paulsen}}}. ]
-    ($K$): For each $(a \otimes \xi), (b \otimes \eta) \in K_0$, let
+    (a): Let $A \otimes H$ and define the Hermitian form
     \[
-    \dpn{a \otimes \xi, b \otimes \eta}{\phi} = \dpn{\phi(b^*a)\xi, \eta}{H}
+    \dpn{\cdot, \cdot}{\phi}: (A \otimes H) \times (A \otimes H)\to \complex \quad \dpn{a \otimes \xi, b \otimes \eta}{\phi} = \dpn{\phi(b^*a)\xi, \eta}{H}
     \]
 
-    and extend by linearity. For each $\seqf{(a_j \otimes \xi_j)} \subset A \otimes H$, let
+    For each $\sum_{j = 1}^n a_j \otimes \xi_j \in A \otimes H$, let $\xi = (\xi_1, \cdots, \xi_n) \in H^n$ and
     \[
-    T = \sum_{i, j = 1}^n E_{ij} \otimes a_i^*a_j = \braks{\sum_{i = 1}^n E_{1i} \otimes a_{i}}^*\braks{\sum_{i = 1}^n E_{1i} \otimes a_{i}} \in M_n(A)
+    a = \sum_{i, j = 1}^n E_{ij} \otimes a_{i}^* a_j = \braks{\sum_{i = 1}^n E_{1i} \otimes a_{i}}^*\braks{\sum_{j = 1}^n E_{1j} \otimes a_j} \ge 0
     \]
 
-    Then $T \ge 0$. As $\phi$ is completely positive, 
+    then since $\phi$ is completely positive, 
     \[
-    \angles{\sum_{j = 1}^n a_j \otimes \xi_j, \sum_{j = 1}^n a_j \otimes \xi_j}_{\phi} = \angles{\phi_n(T)\begin{bmatrix}
+        \angles{\sum_{j = 1}^n a_j \otimes \xi_j, \sum_{j = 1}^n a_j \otimes \xi_j}_\phi = \sum_{i, j = 1}^n \dpn{\phi(a_i^*a_j)\xi_j, \xi_i}{H} = \dpn{\phi_n(a)\xi, \xi}{H^n} \ge 0
-        \xi_1 \\
-        \vdots \\
-        \xi_n
-    \end{bmatrix}, \begin{bmatrix}
-        \xi_1 \\
-        \vdots \\
-        \xi_n
-    \end{bmatrix}}_{H^n} \ge 0
     \]
 
-    so $\dpn{u, u}{\phi}$ is a pseudo inner product. Let
+    and $\dpn{\cdot, \cdot}{\phi}$ is a pseudo-inner product. 
+    
+    Let $N = \bracsn{u \in A \otimes H| \dpn{u, u}{\phi} = 0}$. For any $u \in N$ and $v \in A \otimes H$, by the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}, 
     \[
-    N = \bracsn{u \in A \otimes H| \dpn{u, u}{\phi} = 0}
+    \dpn{u, v}{\phi}| \le \dpn{u, u}{\phi}^{1/2} \dpn{v, v}{\phi}^{1/2} = 0
     \]
 
-    then by the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}, 
+    so $N = \bracsn{u \in A \otimes H|\dpn{u, v}{\phi} = 0 \forall v \in K_0}$. 
+
+    Let $K_0 = (A \otimes H)/N$ and $K$ be the completion of $K_0$ with respect to $\dpn{\cdot, \cdot}{\phi}$, then $K$ is a complex Hilbert space. 
+
+    (b): Let $V: H \to K_0$ be defined by $\xi \mapsto 1_A \otimes \xi + N$, then for each $\xi \in H$, 
     \[
-    N = \bracsn{u \in A \otimes H| \dpn{u, v}{\phi} = 0 \forall v \in A \otimes H}
+    \norm{V\xi}_{K_0}^2 = \dpn{1_A \otimes \xi, 1_A \otimes \xi}{\phi} = \dpn{\phi(1_A)\xi, \xi}{H} \le \norm{\phi(1_A)}_{B(H)}\norm{\xi}_H^2
     \]
 
-    Let $K_0 = (A \otimes H)/N$ and $K$ be the completion of $K_0$, then $K$ is a Hilbert space. 
+    so $\norm{V}_{L(H; K)}^2 \le \norm{\phi(1_A)}_{B(H)}$. 
 
-    ($\pi$): For each $a \in A$, let
+    (c): For each $a \in A$, let
     \[
-    \pi_0(a): K_0 \to K_0 \quad \pi_0(a)(b \otimes \xi + N) = (ab) \otimes \xi + N
+    \pi(a): K_0 \to K_0 \quad \pi(a)(b \otimes \xi + N) = (ab) \otimes \xi + N
     \]
 
-    For any $b \in A$, $b^*a^*ab \le b^*\norm{a}_A^2b = \norm{a}_A^2b^*b$. As such, for any $u = \sum_{j = 1}^n b_j \otimes \xi_j \in  A \otimes H$, 
+    then for any $u = \sum_{j = 1}^n b_j \otimes \xi_j \in A \otimes H$, 
     \begin{align*}
-        \angles{\pi(a)u, \pi(a)u}_{\phi} &= \sum_{i, j = 1}^n \dpn{\pi(a)(b_i \otimes \xi_i), \pi(a)(b_j \otimes \xi_j)}{\phi} \\
+        \norm{\pi(a)(u+N)}_{K_0}^2 &= \sum_{i, j = 1}^n \dpn{(ab_j) \otimes \xi_j, b_i \otimes \xi_i}{\phi} = \sum_{i, j = 1}^n \dpn{\phi(b_i^*a^*ab_j)\xi_j, \xi_i}{H} \\
-        &= \sum_{i, j = 1}^n \dpn{(ab_i) \otimes \xi_i, (ab_j) \otimes \xi_j}{\phi} = \sum_{i, j = 1}^n \dpn{\phi(b_j^*a^*ab_i) \otimes \xi_i,\xi_j}{H} \\
+        &\le \sum_{i, j = 1}^n \norm{a^*a}_A \dpn{\phi(b_i^*b_j)\xi_j, \xi_i}{H} = \norm{a}_A^2 \cdot \norm{u + N}_{K_0}^2
-        &\le \sum_{i, j = 1}^n \norm{a^*a}_{A}\dpn{\phi(b_j^*b_i)\xi_i, \xi_j}{H} = \norm{a}_A^2 \dpn{u, u}{\phi}
     \end{align*}
 
-    Thus $\pi_0(a)$ is well-defined with $\norm{\pi_0(a)}_{B(K_0)} \le \norm{a}_A$, and extends into a continuous linear operator $\pi(a) \in B(K)$. The map $\pi: A \to B(K)$ is then a *-homomorphism. 
+    so $\norm{\pi(a)}_{B(K_0)} \le \norm{a}_A$. By the \hyperref[Linear Extension Theorem]{theorem:linear-extension-theorem-normed}, $\pi(a)$ extends uniquely to an operator in $B(H)$, and $\pi: A \to B(H)$ is a bounded linear map. 
 
-    ($V$): Let $V: H \to K$ be defined by $\xi \mapsto 1_A \otimes \xi + N$. 
+    By definition, $\pi$ is a unital homomorphism of Banach algebras. Finally, for any $a \in A$ and $b \otimes \xi, c \otimes \eta \in A \otimes H$, 
+    \begin{align*}
+        \dpn{\pi(a^*)(b \otimes \xi + N), c \otimes \eta + N}{K_0} &= \dpn{(a^*b) \otimes \xi, c \otimes \eta}{\phi}\\
+        &= \dpn{\phi(c^*a^*b)\xi, \eta}{H} = \dpn{b \otimes \xi, (ac) \otimes \eta}{\phi} \\
+        &= \dpn{b \otimes \xi + N, \pi(a)(c \otimes \eta + N)}{K_0}
+    \end{align*}
+
+    By linearity, $\dpn{\pi(a^*)u, v}{K_0} = \dpn{u, \pi(a)v}{K_0}$ for all $u, v \in K_0$. As $K_0$ is dense in $K$, $\pi(a^*) = \pi(a)^*$, so $\pi$ is a unital *-homomorphism. 
 
     (1): For each $\xi \in H$, 
     \[
-    \dpn{V\xi, V\xi}{K} = \dpn{1_A \otimes \xi, 1_A \otimes \xi}{\phi} = \dpn{\phi(1_A)\xi, \xi}{H}
+    \norm{V\xi}_{K_0}^2 = \dpn{1_A \otimes \xi, 1_A \otimes \xi}{\phi} = \dpn{\phi(1_A)\xi, \xi}{H}
     \]
 
-    so $\normn{\phi(1_A)}_{B(H)} = \norm{V}_{L(H; K)}^2$. In particular, if $\phi$ is unital, then $V$ is an isometry by \hyperref[polarisation]{proposition:polarisation-complex}. 
+    so
+    \[
+    \norm{V}_{K_0}^2 = \sup_{\substack{\xi \in H \\ \norm{\xi}_H \le 1}}\norm{V\xi}_{K_0}^2 = \sup_{\substack{\xi \in H \\ \norm{\xi}_H \le 1}}\dpn{\phi(1_A)\xi, \xi}{H} = w(\phi(1_A)) = \norm{\phi(1_A)}_{B(H)}
+    \]
 
-    (2): Finally, for any $\xi, \eta \in H$, 
+    by \autoref{proposition:bh-numerical-radius-gymnastics}. 
-    \begin{align*}
-        \dpn{V^*\pi(a)V\xi, \eta}{H} &= \dpn{\pi(a)V\xi, V\eta}{K} = \dpn{a \otimes \xi, 1_A \otimes \eta}{\phi} \\
-        &= \dpn{\phi(a)\xi, \eta}{H}
-    \end{align*}
 
+    (2): For each $a \in A$ and $\xi, \eta \in H$, 
+    \[
+        \dpn{V^*\pi(a)V\xi, \eta}{H} = \dpn{\pi(a)V\xi, V\eta}{K} = \dpn{a \otimes \xi, 1_A \otimes \eta}{\phi} = \dpn{\phi(a)\xi, \eta}{H}
+    \]
+
+    so $V^*\pi(a)V = \phi(a)$. 
+
+    (3): If $\phi$ is unital, then for each $\xi \in H$,
+    \[
+    \dpn{V\xi, V\xi}{H} = \dpn{1_A \otimes \xi, 1_A \otimes \xi}{\phi} = \dpn{\phi(1_A)\xi, \xi}{H} = \dpn{\xi, \xi}{H}
+    \]
+
+    so $V$ is an isometry. 
     
 \end{proof}