Generalized additive mapping in Banach modules and isomorphisms between C∗-algebras

Let X, Y be vector spaces. It is shown that if an odd mapping f :X→Y satisfies the functional equation rf d j=1 xj r + ι(j)=0,1 d j=1 ι(j)=l rf d j=1(−1)ι(j)xj r = (d−1Cl −d−1Cl−1 +1) d j=1 f (xj ) (1) then the odd mapping f :X→Y is additive, and we prove the Hyers–Ulam stability of the functional equation (1) in Banach modules over a unital C∗-algebra. As an application, we show that every almost linear bijection h:A→B of a unital C∗-algebra A onto a unital C∗-algebra B is a C∗-algebra isomorphism when h 2n rn uy = h 2n rn u h(y) for all unitaries u ∈ A, all y ∈ A, and n = 0, 1, 2, . . . .  2005 Elsevier Inc. All rights reserved.