RETRACTION: wrong word used.

src/op/system/definitions.tex

@@ -38,12 +38,13 @@
 \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
-    \[
-    \norm{\phi}_{L(S; T)} \le 2\norm{\phi(1_A)}_B
-    \]
+    \begin{enumerate}
+        \item For any $x \in S_{sa}$, $\norm{\phi(x)}_{B} \le \norm{\phi(1_A)}_B \cdot \norom{x}_A$.
+        \item $\norm{\phi}_{L(S; T)} \le 2\norm{\phi(1_A)}_B$. 
+    \end{enumerate}
 \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 $\norm{\phi(x)}_B \le \norm{x}_A \cdot \norm{\phi(1_A)}_B$. 
+    (1): 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, 
     \[
@@ -57,7 +58,7 @@
         &\le \norm{s}_A \norm{\phi(1_A)}_B
     \end{align*}
     
-    Finally, for any general $s \in S$, 
+    (2): 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} \\
@@ -65,6 +66,35 @@
     \end{align*}
 \end{proof}
 
+\begin{theorem}
+\label{theorem:positive-commutative-norm}
+    Let $X$ be a compact Hausdorff space, $A$ be a unital $C^*$-algebra and $\phi: C(X; \complex) \to A$ be a positive map, then $\norm{\phi}_{L(C(X; \complex); B)} = \norm{\phi(\one)}_{B}$. 
+\end{theorem}
+\begin{proof}
+    Assume without loss of generality that $\norm{\phi(\one)}_{B} \le 1$. 
+
+    Let $\seqf{h_j} \subset C(X; [0, 1])$ be a partition of unity on $X$, and $\seqf{\lambda_j} \subset \ol{B_\complex(0, 1)}$. 
+
+    Since $A$ is a unital $C^*$-algebra, there exists a complex Hilbert space $H$ such that $A$ may be identified with a $C^*$-subalgebra of $B(H)$, with $1_A = 1_{B(H)}$. In which case, for any $\xi, \eta \in H$, 
+    \begin{align*}
+        \abs{\sum_{j = 1}^n \dpn{\lambda_j \phi(h_j)\xi, \eta}{H}} &\le \sum_{j = 1}^n \abs{\dpn{\phi(h_j)\xi, \eta}{H}} \\
+        &= \sum_{j = 1}^n\abs{\dpn{\phi(h_j)^{1/2}\xi, \phi(h_j)^{1/2}\eta}{H}} \\
+        &\le \braks{\sum_{j = 1}^n \normn{\phi(h_j)^{1/2}\xi}_H^2}^{1/2} \cdot \braks{\sum_{j = 1}^n\normn{\phi(h_j)^{1/2}\eta}_H^2}^{1/2}\\
+        &\le \braks{\sum_{j = 1}^n \dpn{\phi(h_j)\xi, \xi}{H}}^{1/2} \cdot \braks{\sum_{j = 1}^n\dpn{\phi(h_j)\xi, \xi}{H}}^{1/2} \\
+        &\le \braks{\angles{\sum_{j = 1}^n \phi(h_j)\xi, \xi}{H}}^{1/2} \cdot \braks{\angles{\sum_{j = 1}^n \phi(h_j)\eta, \eta}{H}}^{1/2} \\
+        &\le (\dpn{\xi, \xi}{H})^{1/2}(\dpn{\eta, \eta}{H})^{1/2} =  \norm{\xi}_H \norm{\eta}_H
+    \end{align*}
+
+    so $\normn{\sum_{j = 1}^n \lambda_j\phi(h_j)}_A \le 1$.
+
+    Let $f \in C(X; \complex)$ with $\norm{f}_u \le 1$ and $\eps > 0$, then there exists a partition of unity $\seqf{h_j} \subset C(X; [0, 1])$ such that for each $1 \le j \le n$, $\sup_{x, y \in \supp{h_j}}|f(x) - f(y)| < \eps$. For each $1 \le j \le n$, choose $x_j \in \supp{h_j}$, then $\norm{f - \sum_{j = 1}^n f(x_j)h_j}_u \le \eps$.
+    
+    By the previous case and \autoref{proposition:positive-automatic-continuity}, 
+    \[
+    \norm{\phi(f)}_A \le \norm{\sum_{j = 1}^n f(x_j)\phi(h_j)}_A + 2\norm{f - \sum_{j = 1}^n f(x_j)h_j}_u \le 1 + 2\eps
+    \]
+\end{proof}
+
 
 \begin{example}[Arveson]
     \label{example:positive-automatic-continuity}
@@ -81,7 +111,10 @@
     \begin{enumerate}
         \item $\phi$ is a positive map.
         \item $\norm{\phi}_{L(C(\mathbb{T} ; \complex); M_2(\complex))} = 2$. 
+        \item $\phi$ admits no extension to a positive map on $C(\mathbb{T}; \complex)$. 
     \end{enumerate}
+
+    As such, $M_2(\complex)$ is not injective with respect to operator systems and positive maps. 
 \end{example}
 \begin{proof}[Proof, {{\cite[Example 2.2]{Paulsen}}}. ]
     (1): Let $a + bz + c \ol z \in S$, then the following are equivalent:
@@ -95,4 +128,7 @@
 
     (2): $\norm{\phi(z)}_{M_2(\complex)} = 2$. 
     
+    (3): By \autoref{theorem:positive-commutative-norm}, any such extension would have norm $1$. 
 \end{proof}
+
+