First draft of L1 representations.

src/op/c-star/homomorphism.tex

@@ -17,6 +17,40 @@
 \end{definition}
 
 
+
+\begin{definition}[*-Representation]
+\label{definition:star-rep}
+    Let $A$ be an involutive Banach algebra, then a \textbf{*-representation} of $A$ is a pair $(H, \pi)$, where $H$ is a complex Hilbert space, and $\pi: A \to B(H)$ is a *-homomorphism. 
+\end{definition}
+
+
+\begin{definition}[Non-Degenerate]
+\label{definition:star-rep-nondegen}
+    Let $A$ be an involutive Banach algebra and $(H, \pi)$ be a *-representation of $A$, then the following are equivalent:
+    \begin{enumerate}
+        \item $\text{span}\bracsn{\pi(x)\xi|x \in A, \xi \in H}$ is dense in $H$. 
+        \item There exists no non-zero $\xi \in H$ such that $\pi(x)\xi = 0$ for all $x \in A$.
+    \end{enumerate}
+    
+    If the above holds, then $(H, \pi)$ is \textbf{non-degenerate}.
+\end{definition}
+\begin{proof}
+    $\neg (1) \Rightarrow \neg (2)$: Let $\xi \in \bracsn{\pi(x)\eta|x \in A, \eta \in H}^\perp$, then for any $x \in A$ and $\eta \in H$, 
+    \[
+    \dpn{\pi(x)\xi, \eta}{H} = \dpn{\xi, \pi(x^*)\eta}{H} = 0
+    \]
+
+    so $\pi(x)\xi = 0$ for all $x \in A$. 
+    
+    $\neg (2) \Rightarrow \neg (1)$: Let $\xi \in H \setminus \bracsn{0}$ with $\pi(x)\xi = 0$ for all $x \in A$, then for any $x \in A$ and $\eta \in H$, 
+    \[
+    \dpn{\pi(x)\eta, \xi}{H} = \dpn{\eta, \pi(x)\xi}{H} = 0
+    \]
+
+    so $\bracsn{\pi(x)\eta|x \in A, \eta \in H} \subset \bracsn{\xi}^\perp$, and cannot be dense in $H$.
+\end{proof}
+
+
 \begin{proposition}
 \label{proposition:star-homomorphism-contractive}
     Let $A, B$ be unital $C^*$-algebras and $\phi: A \to B$ be a unital *-homomorphism, then for each $x \in A$,