Definition 38.5.3 (Non-Degenerate).label Let $A$ be an involutive Banach algebra and $(H, \pi)$ be a *-representation of $A$, then the following are equivalent:
- (1)
$\text{span}\bracsn{\pi(x)\xi|x \in A, \xi \in H}$ is dense in $H$.
- (2)
There exists no non-zero $\xi \in H$ such that $\pi(x)\xi = 0$ for all $x \in A$.
If the above holds, then $(H, \pi)$ is non-degenerate.
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 $\eta \in H$,
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 $\eta \in H$,
so $\bracsn{\pi(x)\eta|x \in A, \eta \in H}\subset \bracsn{\xi}^{\perp}$, and cannot be dense in $H$.$\square$
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