Does pointwise nilpotency imply global nilpotency?

Question

Is there a compact Haussdorf space $X$ and $C^{*}$ algebra $A$ with a continuous map $f:X\to A$, such that $f(x)\in A$ is a nilpotent element, $\forall x \in X$, but $f$ is not a nilpotent element of the algebra $C(X,A)$, the space of $A$-valued continuous maps on $A$?

Answer 1

Sure. Take $A=B(H)$ for $H$ infinite-dimensional (or any other algebra that has nilpotent elements of arbitrarily large degree), and choose a sequence of elements $a_n\in A$ such that $a_n^n=0$, $a_n^{n-1}\neq 0$, and $\|a_n\|=1/n$. Taking $X=\mathbb{N}\cup\{\infty\}$, there is then $f\in C(X,A)$ given by $f(n)=a_n$ and $f(\infty)=0$. Each value of $f$ is nilpotent, but there is no single $n$ such that $f(x)^n=0$ for all $x$.