Isomorphisms in unital $C^*$-algebras


1 Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea

2 Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 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
 $h(3^n u y) = h(3^n u) h(y)$ 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  $h(3^n u y) =
h(3^n u) h(y)$  for all
  $u \in \{ v \in A \mid v = v^*, \|v\|=1, v \text{ is invertible} \}$, all
$y \in A$, and all $n\in \mathbb Z$.

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 $T(0) =0$ 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