  Let $\alpha$ be an approximately inner flow on a $C^*$ algebra $A$ with
generator $\delta$ and let $\delta_n$ denote the bounded generators of the
approximating flows $\alpha^{(n)}$. We analyze the structure of the set
  \cd=\{x\in D(\delta): \lim_{n\to\infty}\delta_n(x)=\delta(x)\} of pointwise
convergence of the generators. In particular we examine the relationship of
$\cd$ and various cores related to spectral subspaces.
