- - - Department of Mathematics, Hanyang University, Seoul ۱۳۳-۷۹۱, Republic of Korea
- - - Department of Mathematics, National Technical University of Athens, Zografou Campus, ۱۵۷۸۰ Athens, Greece

It is shown that every  almost linear bijection h : A\rightarrow B of a unital C^*-algebra A onto a unital C^*-algebra B is a C^*-algebra isomorphism when  h(۳^n u y) = h(۳^n u)  h(y) for all unitaries  u \in A, all y \in A, and all n\in \mathbb Z, and that almost linear continuous bijection h : A \rightarrow B of a unital C^*-algebra A of real rank zero onto a unital C^*-algebra B is a C^*-algebra isomorphism when  h(۳^n u y) = h(۳^n u) h(y)  for  all   u \in \{ v \in A \mid v = v^*, \|v\|=۱, v \text{ is invertible} \}, all y \in A, and all  n\in \mathbb Z. Assume that X and Y  are left normed modules over a unital C^*-algebra  A. It is shown that every surjective isometry T : X \rightarrow Y, satisfying T(۰) =۰ and T(ux) = u T(x) for all x \in X and all unitaries u \in A, is an A-linear isomorphism. This is applied to investigate C^*-algebra isomorphisms in unital C^*-algebras.

generalized Hyers-Ulam stability, C^*-algebra isomorphism, real rank zero, isometry