On the generalized Hyers–Ulam stability of module left derivations

Let A be a unital normed algebra and letMbe a unitary Banach left A-module. If f : A→Mis an approximate module left derivation, then f : A→M is a module left derivation. Moreover, if M= A is a semiprime unital Banach algebra and f (tx) is continuous in t ∈ R for each fixed x in A, then every approximately linear left derivation f : A→A is a linear derivation which maps A into the intersection of its center Z(A) and its Jacobson radical rad(A). In particular, if A is semisimple, then f is identically zero. © 2007 Elsevier Inc. All rights reserved