Isomorphisms in unital $C^*$-algebras

It is shown that every  almost linear bijection $h : Arightarrow B$ of a unital $C^*$-algebra $A$ onto a unital$C^*$-algebra $B$ is a $C^*$-algebra isomorphism when $h(3^n u y) = h(3^n u) h(y)$ for allunitaries  $u in A$, all $y in A$, and all $nin mathbb Z$, andthat almost linear continuous bijection $h : A rightarrow B$ of aunital $C^*$-algebra $A$ of real rank zero onto a unital$C^*$-algebra $B$ is a $C^*$-algebra isomorphism when  $h(3^n u y) =h(3^n u) h(y)$  for all  $u in { v in A mid v = v^*, |v|=1, v text{ is invertible} }$, all$y in A$, and all $nin mathbb Z$.Assume that $X$ and $Y$  are left normed modules over a unital$C^*$-algebra $A$. It is shown that every surjective isometry $T : Xrightarrow Y$, satisfying $T(0) =0$ and $T(ux) = u T(x)$ for all $xin X$ and all unitaries $u in A$, is an $A$-linear isomorphism.This is applied to investigate $C^*$-algebra isomorphisms in unital$C^*$-algebras.