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 26739 \mathcal D=\bigl\{x\in D(\delta): \lim_{n\rightarrow\infty}\delta_n(x)=\delta(x)\bigr\} 26739 of pointwise convergence of the generators. In particular we examine the relationship of $\mathcal D$ and various cores related to spectral subspaces.
