Consolidating.

src/op/system/definitions.tex

@@ -1,9 +1,98 @@
 \section{Operator Systems}
 \label{section:operator-systems}
 
-\begin{definition}[Operator System]
-\label{definition:operator-system}
-    Let $A$ be a $C^*$-algebra and $S \subset A$, then $S$ is an \textbf{operator system} if $S$ is a linear subspace of $A$ and $S = S^*$. 
+\begin{definition}[Concrete Operator System]
+\label{definition:concrete-operator-system}
+    Let $A$ be a $C^*$-algebra and $S \subset A$, then $S$ is a \textbf{concrete operator system} if:
+    \begin{enumerate}
+        \item $1_A \in S$.
+        \item $S = S^*$. 
+        \item $S$ is a linear subspace of $A$. 
+    \end{enumerate}
+\end{definition}
+
+\begin{definition}[Self-Adjoint Part]
+\label{definition:operator-system-sa}
+    Let $A$ be a $C^*$-algebra and $S \subset A$ be an operator system, then $S_{sa} = \bracsn{x \in S|x = x^*}$ is the \textbf{self-adjoint part} of $S$. 
+\end{definition}
+
+\begin{definition}[Positive Cone]
+\label{definition:operator-system-positive-cone}
+    Let $A$ be a $C^*$-algebra and $S \subset A$ be an operator system, then $S_{+} = \bracsn{x \in S_{sa}|x \ge 0}$ is the \textbf{positive cone} of $S$. 
 \end{definition}
 
 
+\begin{proposition}
+\label{proposition:operator-system-positive-span}
+    Let $A$ be a $C^*$-algebra, $S \subset A$ be an operator system, and $x \in S_{sa}$, then there exists positive elements $p, q \in S_{sa}$ such that $x = p - q$. 
+\end{proposition}
+\begin{proof}
+    Let $p = (\norm{x}_A + x)/2$ and $q = (\norm{x}_A - x)/2$, then $q$ is positive by \autoref{corollary:spectrum-characterisation-iff}. 
+\end{proof}
+
+\begin{definition}[Positive]
+\label{definition:operator-system-positive-map}
+    Let $A, B$ be $C^*$-algebra, $S \subset A$ and $T \subset B$ be operator systems, and $\phi: S \to T$ be a linear map, then $\phi$ is \textbf{positive} if $\phi(S_+) \subset \phi(T_+)$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:positive-automatic-continuity}
+    Let $A, B$ be $C^*$-algebra, $S \subset A$ and $T \subset B$ be operator systems, and $\phi: S \to T$ be a positive map, then
+    \[
+    \norm{\phi}_{L(S; T)} \le 2\norm{\phi(1_A)}_B
+    \]
+\end{proposition}
+\begin{proof}
+    Let $x \in T_+$, then $0 \le x \le \norm{x}_A$, and $0 \le \phi(x) \le \norm{x}_A \cdot \norm{\phi(1_A)}_B$, so $\norm{\phi(x)}_B \le \norm{x}_A \cdot \norm{\phi(1_A)}_B$. 
+
+    For any $x, y \in T_+$, $x - y \le x$ and $y - x \le y$. As such, 
+    \[
+    \norm{x - y}_{B} \le \norm{x}_B \vee \norm{y}_B
+    \]
+
+    In particular, for any $s \in S_{sa}$, 
+    \begin{align*}
+        \phi(s) &= \frac{1}{2}\braks{\phi(\norm{s}_A + s) - \phi(\norm{s}_A - s)} \\
+        \norm{\phi(s)}_{B} &\le \frac{1}{2}\braks{\norm{\phi(\norm{s}_A + s)}_B \vee \norm{\phi(\norm{s}_A - s)}_B} \\
+        &\le \norm{s}_A \norm{\phi(1_A)}_B
+    \end{align*}
+    
+    Finally, for any general $s \in S$, 
+    \begin{align*}
+        \phi(s) &= \phi(\text{Re}(s)) + i\phi(\text{Im}(s)) \\
+        \norm{\phi(s)}_B &\le \norm{\phi(\text{Re}(s))}_B + \norm{\phi(\text{Im}(s))}_{B} \\
+        &\le \norm{\phi(1_A)}_B \cdot (\norm{\text{Re}(s)}_A + \norm{\text{Im}(s)}_A) \le 2\norm{\phi(1_A)}_B\norm{s}_A
+    \end{align*}
+\end{proof}
+
+
+\begin{example}[Arveson]
+    \label{example:positive-automatic-continuity}
+
+    Let $\mathbb{T} = \bracsn{z \in \complex|\ |z| = 1}$, and $S = \text{span}\bracsn{1, z, \ol z}  \subset C(\mathbb{T}; \complex)$, then $S$ is an operator system in $C(\mathbb{T} ; \complex)$. Let
+    \[
+    \phi: S \to M_2(\complex) \quad (a + bz + c\ol z) = \begin{bmatrix}
+        a & 2b \\
+        2c & a
+    \end{bmatrix}
+    \]
+
+    then:
+    \begin{enumerate}
+        \item $\phi$ is a positive map.
+        \item $\norm{\phi}_{L(C(\mathbb{T} ; \complex); M_2(\complex))} = 2$. 
+    \end{enumerate}
+\end{example}
+\begin{proof}[Proof, {{\cite[Example 2.2]{Paulsen}}}. ]
+    (1): Let $a + bz + c \ol z \in S$, then the following are equivalent:
+    \begin{enumerate}
+        \item $a + bz + c\ol z \ge 0$. 
+        \item $c = \ol b$ and $a \ge 2\text{Re}(b)$. 
+        \item $c = \ol b$ and $a \ge 2|b|$. 
+    \end{enumerate}
+
+    In addition, since a $2 \times 2$ matrix is positive if and only if its diagonal entries and its determinant are non-negative. Thus $\phi$ is positive. 
+
+    (2): $\norm{\phi(z)}_{M_2(\complex)} = 2$. 
+    
+\end{proof}