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

Abstract

Let A be a ring and σ : A → A a ring endomorphism. A generalized skew (or σ-)derivation of A is an additive map d : A → A for which there exists a map δ :algebra and σ is surjective, we determine the structure of generalized σ-derivations of A that belong to the cb-norm closure of elementary operators Eℓ(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 σ : A → A lies in the cb-norm closure of Eℓ(A), then σ 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, M 2 ).