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 $C^*$-algebras.