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

Let X,YX,Y be vector spaces. It is shown that if an odd mapping f:X→Y satisfies the functional equationequation(1)rf(∑j=1dxjr)+∑ι(j)=0,1∑j=1dι(j)=lrf(∑j=1d(−1)ι(j)xjr)=(Cld−1−Cl−1d−1+1)∑j=1df(xj) 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∗C∗-algebra. As an application, we show that every almost linear bijection h:A→B of a unital C∗C∗-algebra A   onto a unital C∗C∗-algebra B   is a C∗C∗-algebra isomorphism whenh(2nrnuy)=h(2nrnu)h(y) for all unitaries u∈Au∈A, all y∈Ay∈A, and n=0,1,2,….