On the Cauchy–Rassias stability of the Trif functional equation in C∗-algebras

Let A, B be two unital C∗-algebras, and let q := k(n−1)/(n− k) for given integers k,n with 2 k n − 1. Consider an almost unital approximately linear mapping h:A→B. We prove that h is a homomorphism when h(q−j xu) = h(x)h(q−j u) for all x ∈ A, all unitaries u ∈ A, and all sufficiently large integers j. Moreover, when A has real rank zero, we give conditions in order for h to be a ∗-homomorphism. Furthermore, we investigate the Cauchy–Rassias stability of the Trif functional equation associated with ∗-homomorphisms between unital C∗-algebras and ∗-derivations of a unital C∗-algebra.  2004 Elsevier Inc. All rights reserved.