Let $A$ be a $C^*$-Algebra and $a\in A$. Assume that $p(a)=q(a)$ for all formal complex polynomials $p(X),q(X)$ that coincide as functions on the spectrum of $a$. Does this imply that $a$ is normal?
The converse is certainly true by the continuous functional calculus. In case this is true, I'm also interested if this holds in $*$-Algebras or Banach-$*$-Algebras.
It does not. At the very least, it suffices for $a$ to be similar to a normal matrix. That is, we may have that $a = bcb^{-1}$, where $c$ is normal.
For instance, take $A = \Bbb C^{2 \times 2}$, and $$ a = \pmatrix{2&1\\0&1} $$ Over finite dimensional $C^*$-algebras, $p(a) = q(a)$ for all complex polynomials that coincide on the spectrum will imply that $a$ is diagonalizable, which is to say that $a$ is similar to some normal element. I am not sure whether the same will hold true in general.