C*-algebras: Existence of an element inducing an injective map

Question

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?

Answer 1

In any Banach space $B$, if $S$ is a countable set of bounded linear maps, there is $a \in B$ such that $S\ni T \mapsto Ta \in B$ is injective. This follows easily from the Baire Category Theorem: $B$ is not the union of the countably many closed, nowhere dense sets $\{x \in B \mid T_i x = T_j x\}$.