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

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 $C^{\ast }$-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 $C^{\ast }$-algebras like $C(X,{\mathbb{M}}_2)$.

13 pages, to appear in Linear and Multilinear Algebra

Cited by in corpus (1)

• Elementary operators on Hilbert modules over prime $C^{\ast }$-algebras