Added GNS.

src/op/c-star/state.tex

@@ -9,8 +9,8 @@
     The set of states $S(A) \subset A^*$ of $A$ equipped with the weak* topology is the \textbf{state space} of $A$.
 \end{definition}
 
-\begin{lemma}
-\label{lemma:cstar-state-cauchy-schwarz}
+\begin{definition}
+\label{definition:cstar-state-pseudo-inner-product}
     Let $A$ be a unital $C^*$-algebra and $\phi \in A^*$ be a positive linear functional, then the mapping
     \[
     A \times A \to \complex \quad (x, y) \mapsto \dpn{x, y}{\phi} := \dpn{y^*x, \phi}{A}
@@ -20,7 +20,9 @@
     \[
     |\dpn{y^*x, \phi}{A}|^2 = |\dpn{x, y}{\phi}|^2 \le \dpn{x, x}{\phi} \cdot \dpn{y, y}{\phi}
     \]
-\end{lemma}
+
+    The pairing $\dpn{\cdot, \cdot}{\phi}$ is the \textbf{pseudo inner product associated with $\phi$}. 
+\end{definition}
 \begin{proof}
     By the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}. 
 \end{proof}
@@ -60,7 +62,7 @@
 \begin{proof}
     (1): Let $\phi \in \Omega(A)$. By \autoref{proposition:multiplicative-unit}, $\norm{\phi}_{A^*} = \dpn{1, \phi}{A} = 1$. Thus $\phi$ is a state by \autoref{theorem:cstar-positive-algebraic}, and $\Omega(A) \subset S(A)$. 
 
-    Let $\psi, \rho \in S(A)$ and $t \in (0, 1)$ such that $\phi = (1 - t)\psi + t\rho$, then for each $x \in \ker(\phi)$, $x^*x \in \ker(\phi)$ as well. As $t \ne 0$, $x^*x \in \ker(\psi)$ and $x^*x \in \ker(\rho)$. By the \hyperref[Cauchy-Schwarz inequality]{lemma:cstar-state-cauchy-schwarz}, 
+    Let $\psi, \rho \in S(A)$ and $t \in (0, 1)$ such that $\phi = (1 - t)\psi + t\rho$, then for each $x \in \ker(\phi)$, $x^*x \in \ker(\phi)$ as well. As $t \ne 0$, $x^*x \in \ker(\psi)$ and $x^*x \in \ker(\rho)$. By the \hyperref[Cauchy-Schwarz inequality]{definition:cstar-state-pseudo-inner-product}, 
     \[
     |\dpn{x, \psi}{A}|^2 = |\dpn{1^*x, \psi}{A}|^2 \le \dpn{1, \psi}{A} \cdot \dpn{x^*x, \psi}{A} = 0
     \]