Let $A$ be a unital Banach algebra.The maximal ideal space of $A$ is denoted by $\hat A$. Assume that $D:A \to A$ is a derivation. Fix an element $a\in A$.

Assume that for every $\phi\in \hat A$ there exist a natural number $n$ such that $\phi (D^n(a))=0$. Does this impliy that $D^k(a)=0$ for some $k$?

One can consider the same question for a $C^*$ algebra $A$ by replacing the maximal ideal space with the space of irreducible representations.

The motivation comes from the following classical fact which I learned it from the book of Boas "A primer of analytic functions"

Fact: Assume that $f:\mathbb{R} \to \mathbb{R}$ is smooth function duch that for every $x\in \mathbb{R}$ there exist a natural number $n$ with $f^{(n)}(x)=0$.Then $f$ is a polynomial.