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

## TL;DR

This paper characterizes generalized skew derivations in prime C*-algebras that are limits of elementary operators, showing they are of a specific form and that certain endomorphisms must be inner automorphisms.

## Contribution

It provides a structural description of cb-norm limits of elementary operators for generalized skew derivations in prime C*-algebras, and characterizes when endomorphisms are inner automorphisms.

## Key findings

- Generalized skew derivations in prime C*-algebras are of the form $d(x)=bx+axc$ for some $a,b,c$ in the multiplier algebra.
- Endomorphisms in the cb-norm closure of elementary operators are necessarily inner automorphisms.
- The results do not extend to certain non-prime C*-algebras like $C(X, M_2)$.

## Abstract

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 $C(X,\mathbb{M}_2 )$.