3
$\begingroup$

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?

$\endgroup$
0
10
$\begingroup$

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 $\{x \in B \mid T_i x = T_j x\}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.