diff --git a/refs.bib b/refs.bib index 2071454..564f46a 100644 --- a/refs.bib +++ b/refs.bib @@ -292,4 +292,12 @@ note = {URL:https://math.stackexchange.com/q/150258 (version: 2012-05-27)}, eprint = {https://math.stackexchange.com/q/150258}, url = {https://math.stackexchange.com/q/150258} +} + +@book{Paulsen, + title={Completely bounded maps and operator algebras}, + author={Paulsen, Vern I and others}, + volume={78}, + year={2002}, + publisher={Cambridge University Press} } \ No newline at end of file diff --git a/src/dyn/top/def.tex b/src/dyn/top/def.tex index 053dc53..511fdb7 100644 --- a/src/dyn/top/def.tex +++ b/src/dyn/top/def.tex @@ -34,14 +34,65 @@ \begin{definition}[Transitive] \label{definition:tds-transitive} Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $x$ is \textbf{topologically transitive} with respect to $G$ if $Gx$ is dense in $X$. + + The flow $G \curvearrowright X$ is \textbf{point transitive} if it admits a topologically transitive point. + + Finally, $G \curvearrowright X$ is \textbf{topologically transitive} if for any $U, V \subset X$ non-empty and open, there exists $g \in G$ such that $gU \cap V \ne \emptyset$\footnote{Approximations of points may be translated into approximations of other points. }. \end{definition} +\begin{proposition} +\label{proposition:tds-point-transitive} + Let $G \curvearrowright X$ be a point transitive $G$-flow, then $G \curvearrowright X$ is also topologically transitive. +\end{proposition} +\begin{proof} + Let $x \in G$ be a topologically transitive point and $U, V \subset X$ be non-empty and open. Since $Gx$ is dense in $G$, $Gx \cap U \ne \emptyset$ and $Gx \cap V \ne \emptyset$. After translating $x$, assume without loss of generality that $x \in U$, then $GU \supset Gx \cap V \ne \emptyset$. Therefore there exists $g \in G$ such that $gU \cap V \ne \emptyset$. +\end{proof} + \begin{definition}[Periodic] \label{definition:tds-periodic} Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $x$ is \textbf{periodic} with respect to $G$ if $Gx$ is finite. \end{definition} +\begin{definition}[Minimality] +\label{definition:tds-minimal} + Let $G \curvearrowright X$ be a $G$-flow, then the following are equivalent: + \begin{enumerate} + \item $G \curvearrowright X$ admits no proper subflows. + \item Every point in $X$ is topologically transitive. + \end{enumerate} + + + If the above holds, then $G \curvearrowright X$ is \textbf{minimal}. +\end{definition} +\begin{proof} + (1) $\Rightarrow$ (2): Let $x \in X$, then $\ol{Gx}$ with the restricted action is a $G$-subflow of $X$. As $\emptyset \ne \ol{Gx}$ and $G \curvearrowright X$ admits no proper subflows, $\ol{Gx} = x$. + + (2) $\Rightarrow$ (1): Let $G \curvearrowright Y$ be a $G$-subflow of $G \curvearrowright X$. For any $x \in Y$, $Y \supset \ol{Gx} = X$, so $Y = X$. +\end{proof} +\begin{definition}[Bernoulli Shift] +\label{definition:bernoulli-shift} + Let $G$ be a group, $X = 2^G$ equipped with the product topology, and + \[ + \alpha: X \to X \quad \alpha_g(x)(h) = x(g^{-1}h) + \] + + Viewing $2^G$ as the power set of $G$, $\alpha_g(A) = gA$ for any $A \subset G$. The $G$-flow $G \curvearrowright X$ is the \textbf{Bernoulli shift} of $G$, and $G$-subflows of $X$ are \textbf{subshifts} of $G$. +\end{definition} + +\begin{example}[The Shift] + \label{example:bernoulli-shift-z} + + Let $\integer \curvearrowright 2^\integer$ be the Bernoulli shift of $G$, then: + \begin{enumerate} + \item For any $x, y \in 2^\integer$, $y \in \ol{\integer x}$ if and only if for each $A \subset \integer$, there exists $n \in \integer$ such that $y|_A = nx|_A$. In other words, $x$ contains a copy of any segment of $y$\footnote{Topological conditions in Bernoulli shifts translate into combinatorial information. }. + \item $\integer \curvearrowright 2^\integer$ is point transitive, and hence topologically transitive\footnote{It is easier to show topological transitivity: move the trees until they are disjoint}. + \item + \end{enumerate} + +\end{example} + + diff --git a/src/logic/index.tex b/src/logic/index.tex index 326a1db..91a6205 100644 --- a/src/logic/index.tex +++ b/src/logic/index.tex @@ -2,3 +2,6 @@ \label{part:logic} This section is plagiarised from + +\input{./tp/index.tex} +\input{./notation.tex} diff --git a/src/logic/notation.tex b/src/logic/notation.tex new file mode 100644 index 0000000..84f89c5 --- /dev/null +++ b/src/logic/notation.tex @@ -0,0 +1,11 @@ +\chapter{Notations} +\label{chap:logic-notations} + +\begin{tabular}{lll} + \textbf{Notation} & \textbf{Description} & \textbf{Source} \\ + \hline + % ---- Topological Vector Spaces ---- + $\mathcal{L} = (\mathcal{F}, \mathcal{R}, \mathcal{C})$ & Language $\mathcal{L}$ with function symbols $\mathcal{F}$, relation symbols $\mathcal{R}$, and constant symbols $\mathcal{C}$. & \autoref{definition:language} \\ + $\mathcal{M} = (M, f^\cm, R^\cm, c^\cm)$ & $\mathcal{L}$-structure with functions $\bracsn{f^\cm|f \in \mathcal{F}}$, relations $\bracsn{R^\cm|R \in \mathcal{R}}$, and constants $\bracsn{c^\cm|c \in \mathcal{C}}$. & \autoref{definition:language-structure} +\end{tabular} + diff --git a/src/logic/tp/lang.tex b/src/logic/tp/lang.tex index 5ae0778..38acfb0 100644 --- a/src/logic/tp/lang.tex +++ b/src/logic/tp/lang.tex @@ -11,5 +11,46 @@ \end{enumerate} \end{definition} +\begin{definition}[$\mathcal{L}$-Structure] +\label{definition:language-structure} + Let $L = (\mathcal{F}, \mathcal{R}, \mathcal{C})$ be a language, then a \textbf{\mathcal{L}-structure} $\mathcal{M}$ is a family + \[ + \mathcal{M} = (M, f^\cm, R^\cm, c^\cm| f \in \mathcal{F}, R \in \mathcal{R}, c \in \mathcal{C}) + \] + + where + + \begin{enumerate} + \item $M$ is a non-empty set, known as the \textbf{universe}, \textbf{domain}, or \textbf{underlying set} of $\mathcal{M}$. + \item For each $f \in \mathcal{F}$, $f^\cm: M^{n_f} \to M$ is a function in $n$-variables in $M$. + \item For each $R \in \mathcal{R}$, $R^\cm \subset M^{n_R}$ is a $n$-ary relation. + \item For each $c \in \mathcal{C}$, $c^\cm \in M$ is a constant. + \end{enumerate} + + The assignments $f^\cm$, $R^\cm$, and $c^\cm$ are the \textbf{interpretations} of $(\mathcal{F}, \mathcal{R}, \mathcal{C})$. +\end{definition} + +\begin{definition}[Terms] +\label{definition:logic-terms} + Let $\mathcal{L} = (\mathcal{F}, \mathcal{R}, \mathcal{C})$ be a language, then the set of \textbf{$\mathcal{L}$-terms} is the smallest set of expressions $\mathcal{T}(\mathcal{L})$ such that: + \begin{enumerate} + \item $\mathcal{C} \subset \mathcal{T}(\mathcal{L})$. + \item For each $n \in \natp$, the variable symbol $v_n \in \mathcal{T}(\mathcal{L})$. + \item For each $f \in \mathcal{F}$ and $\bracsn{t_j}_1^{n_f} \subset \mathcal{T}(\mathcal{L})$, the expression $f(t_1, \cdots, t_{n_f}) \in \mathcal{T}(\mathcal{L})$. + \end{enumerate} +\end{definition} + +\begin{lemma} +\label{lemma:logic-terms-induction} + Let $\mathcal{L} = (\mathcal{F}, \mathcal{R}, \mathcal{C})$ be a language and $\mathcal{T}_0(\mathcal{L}) = \mathcal{C} \cup \seq{v_n}$. For each $n \in \natp$, inductively define + \[ + \mathcal{T}_{n}(\mathcal{L}) = \mathcal{T}_{n-1}(\mathcal{L}) \cup \bracsn{f(t_1, \cdots, t_{n_f})|f \in \mathcal{F}, \bracsn{t_j}_1^{n_f} \in \mathcal{T}_{n-1}(\mathcal{L})} + \] + + then $\mathcal{T} = \bigcup_{n \in \natz}\mathcal{T}_n$. For each $n \in \natz$, $\mathcal{T}_n$ is the set of \textbf{$\mathcal{L}$-terms of complexity at most $n$}. +\end{lemma} + + + diff --git a/src/op/example/bounded.tex b/src/op/example/bounded.tex index 50adb48..f01ad7e 100644 --- a/src/op/example/bounded.tex +++ b/src/op/example/bounded.tex @@ -81,3 +81,123 @@ \end{proof} +\begin{definition}[Unitary Dilation] +\label{definition:unitary-dilation} + Let $H$ be a complex Hilbert space, $V \in B(H)$ be isometric, $P = I_H - VV^*$ be the projection onto $V(H)^\perp$, and + \[ + U = \begin{bmatrix} + V & P \\ + 0 & V^* + \end{bmatrix} + \] + + then: + \begin{enumerate} + \item $U^*U = UU^* = I_{H^2}$. In other words, $U$ is an isometry on $H^2$. + \item Under the identification that $H = H \times \bracsn{0} \subset H^2$, let $Q \in B(H^2)$ be the projection onto $H$, then $V^n = QU^n|_H$ for all $n \ge 0$. + \end{enumerate} + + The operator $U \in B(H^2)$ is a \textbf{unitary dilation} of $V$. +\end{definition} +\begin{proof} + (1): By direct computation, + \begin{align*} + UU^* &= \begin{bmatrix} + V & P \\ + 0 & V^* + \end{bmatrix} \cdot + \begin{bmatrix} + V^* & 0 \\ + P & V + \end{bmatrix} = \begin{bmatrix} + VV^* + P & PV \\ + V^*P & V^*V + \end{bmatrix} = I_{H^2} \\ + U^*U &= \begin{bmatrix} + V^* & 0 \\ + P & V + \end{bmatrix} \cdot + \begin{bmatrix} + V & P \\ + 0 & V^* + \end{bmatrix} + = \begin{bmatrix} + V^*V & V^*P \\ + PV & VV^* + P + \end{bmatrix} = I_{H^2} \\ + \end{align*} + + + (2): For each $n \in \natp$, + \[ + U^n = \begin{bmatrix} + V & P \\ + 0 & V^* + \end{bmatrix}^n = \begin{bmatrix} + V^n & \cdots \\ + 0 & (V^*)^n + \end{bmatrix} + \] + + so $QU^n|_{H} = V^n$. +\end{proof} + +\begin{definition}[Isometric Dilation] +\label{definition:isometric-dilation} + Let $H$ be a complex Hilbert space, $T \in B(H)$ with $\norm{T}_{B(H)} \le 1$, $D_T = (1 - T^*T)^{1/2}$, and + \[ + V: l^2(\natp; H) \to l^2(\natp; H) \quad (Vx)_n = \begin{cases} + Tx_1 &n = 1 \\ + D_Tx_1 &n = 2 \\ + x_{n-1} &n \ge 3 + \end{cases} + \] + + then: + \begin{enumerate} + \item $V$ is isometric on $l^2(\natp; H)$. + \item Under the identification that $H = H \oplus 0 \oplus \cdots$, let $Q$ be the orthogonal projection onto $H$, then $T^n = QV^n|_H$ for all $n \ge 0$. + \end{enumerate} + + The operator $V$ is an \textbf{isometric dilation} of $T$. +\end{definition} +\begin{proof} + (1): Since $\norm{T}_H \le 1$, $1 - T^*T \ge 0$, and $(1 - T^*T)^{1/2}$ is well-defined. For each $x \in H$, + \begin{align*} + \norm{Tx}_H^2 + \norm{D_Tx}_H^2 &= \dpn{T^*Tx, x}{H} + \dpn{(1 - T^*T)x, x}{H} \\ + &= \dpn{x, x}{H} = \norm{x}_H^2 + \end{align*} + + Thus for each $x = \seq{x_n} \in l^2(H)$, + \begin{align*} + \norm{Vx}_{l^2(\natp; H)}^2 &= \norm{Tx_1}_H^2 + \norm{D_Tx_1}_H^2 + \sum_{n = 3}^\infty \norm{x_{n-1}}_H^2 \\ + &= \norm{x}_H^2 + \sum_{n = 2}^\infty \norm{x_{n}}_H^2 = \norm{x}_{l^2(\natp; H)}^2 + \end{align*} +\end{proof} + +\begin{theorem}[Sz.-Nagy's Dilation Theorem] +\label{theorem:sznagy-dilation} + Let $H$ be a complex Hilbert space and $T \in B(H)$ with $\norm{T}_{B(H)} \le 1$, then there exists a Hilbert space $K \supset H$ and a unitary operator $U \in B(K)$ such that $T^n = P_HU^n|_H$ for all $n \in \natp$, where $P_H$ is the orthogonal projection onto $H$. +\end{theorem} +\begin{proof}[Proof, {{\cite[Theorem 1.1]{Paulsen}}}. ] + Let $K = l^2(\natp; H)^2$ and identify $H = (H \oplus \bracsn{0} \oplus \cdot) \times \bracsn{0}$. Let $V \in B(l^2(\natp; H))$ be the isometric dilation of $T$, and $U \in B(l^2(\natp; H)^2)$ be the unitary dilation of $V$. Identify $l^2(\natp; H)$ as a subspace of $K$ as $l^2(\natp; H) \times \bracsn{0}$, and let $P_{l^2(\natp; H)}$ be the orthogonal projection onto $l^2(\natp; H)$, then for each $n \ge 0$, + \begin{align*} + V^n &= P_{l^2(\natp; H)}U^n|_{l^2(\natp; H)} \\ + T^n = P_{H}V^n|_{H} &= P_HU^n|_{H} + \end{align*} +\end{proof} + +\begin{corollary}[Von Neumann's Inequality] +\label{corollary:von-neumann-inequality-polynomial} + Let $H$ be a complex Hilbert space and $T \in B(H)$ with $\norm{T}_{B(H)} \le 1$, then for any $p \in \complex[x]$, + \[ + \norm{p(T)}_{B(H)} \le \sup_{z \in B_\complex(0, 1)}|p(z)| + \] +\end{corollary} +\begin{proof}[Proof, {{\cite[Corollary 1.2]{Paulsen}}}. ] + By \hyperref[Sz.-Nagy's Dilation Theorem]{theorem:sznagy-dilation}, there exists a Hilbert space $K \supset H$ and a unitary operator $U \in B(K)$ such that $T^n = P_HU^n|_H$ for all $n \in \natp$, where $P_H$ is the orthogonal projection onto $H$. + + Thus for any $p \in \complex[x]$, $p(T) = P_Hp(U)|_H$. As $U$ is unitary, the \hyperref[continuous functional calculus]{definition:continuous-functional-calculus} shows that $\norm{p(U)}_{B(K)} \le \sup_{z \in B_\complex(0, 1)}|p(z)|$. Therefore $\norm{p(T)}_{B(H)} \le \sup_{z \in B_\complex(0, 1)}|p(z)|$. +\end{proof} + + diff --git a/src/op/example/matrix.tex b/src/op/example/matrix.tex index 16cb09a..ca0a8e1 100644 --- a/src/op/example/matrix.tex +++ b/src/op/example/matrix.tex @@ -82,5 +82,3 @@ \] \end{example} - -