A normal element in a Banach $*$-algebra with real spectrum but not self-adjoint

Question

Let $A$ be a Banach $*$-algebra. Let $a\in A$ be a normal element, i.e. $a^*a=aa^*$.

Question: If the spectrum of $a$ is real, must $a$ be self-adjoint?

Thoughts: If $A$ is a $C^*$-algebra, then it is a standard fact that the answer is yes and follows from Gelfand duality. However, for a general Banach $*$-algebra, the Gelfand map is $*$-preserving if and only if every self-adjoint element has real spectrum, and there are examples where $a$ is self-adjoint but does not have real spectrum. I'm curious about the converse.

Answer 1

Consider $A=\mathbb{C}[x]/(x^2)$, with the involution $*$ given by conjugation on $\mathbb{C}$ and $x\mapsto -x$. Since $A$ is commutative, every element of $A$ is normal. The spectrum of the element $x\in A$ is just $\{0\}$, but $x$ is not self-adjoint.