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Let $A$ be a commutative 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 ofR.P Boas "A primer of real 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.

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    $\begingroup$ In the first part of your question, with a general Banach algebra, are you assuming it is commutative? Because if not, the maximal ideal space could be empty. $\endgroup$
    – Yemon Choi
    Commented Apr 22, 2019 at 20:59
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    $\begingroup$ In your second paragraph, you are trying to deduce behaviour of the iterates of $D$ from information defined in terms of the characters of $A$. What do you intend to do in cases where $A$ has a large Jacobson radical (which in the commutative case is the intersection of the kernels of characters) $\endgroup$
    – Yemon Choi
    Commented Apr 22, 2019 at 21:00
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    $\begingroup$ Finally: as soon as you are considering derivations from a commutative Banach algebra to itself, you should always have in mind the Singer-Wermer theorem (this can be found in the book of Bonsall-Duncan, or in the book of Dales) $\endgroup$
    – Yemon Choi
    Commented Apr 22, 2019 at 21:02
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    $\begingroup$ It might be more interesting to consider a densely defined derivation. $\endgroup$
    – Ruy
    Commented Apr 22, 2019 at 23:39
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    $\begingroup$ @YemonChoi So I understand my question for commutative case is trivial since the range of $D$ is contained in the kernel of all character. Thanks for your help. $\endgroup$
    – Ali Taghavi
    Commented Apr 23, 2019 at 11:00

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