The cb-norm approximation of generalized skew derivations by elementary operators

Let $A$ be a ring and $\sigma: A \to A$ a ring endomorphism. A generalized skew (or $\sigma$-)derivation of $A$ is an additive map $d: A \to A$ for which there exists a map $\delta:A \to A$ such that $d(xy)=\delta(x)y+\sigma(x)d(y)$ for all $x,y \in A$. If $A$ is a prime $C^*$-algebra and $\sigma$ is surjective, we determine the structure of generalized $\sigma$-derivations of $A$ that belong to the cb-norm closure of elementary operators $\mathcal{E}\ell(A)$ on $A$; all such maps are of the form $d(x)=bx+axc$ for suitable elements $a,b,c$ of the multiplier algebra $M(A)$. As a consequence, if an epimorphism $\sigma: A \to A$ lies in the cb-norm closure of $\mathcal{E}\ell(A)$, then $\sigma$ must be an inner automorphism. We also show that these results cannot be extended even to relatively well-behaved non-prime $C^*$-algebras like $C(X,\mathbb{M}_2 )$.