I'm wondering if the following statement is true: Let $A$ be a $C^*$-algebra and $\phi: A\rightarrow A$ be a $*$-homomorphism. Is there always an element $a\in A$ such that the map

$\left\{ \phi^{n}:=\phi \circ ...\circ \phi \mid n\in\mathbb{N}\right\} \rightarrow A$, $\phi^n\mapsto \phi^{n}\left(a\right)$

is injective?