Document Type : Research Paper

Authors

• C. Park ¹
• Th. M. Rassias ²

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

Abstract

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

Keywords

• generalized Hyers-Ulam stability
• $C^{\ast }$-algebra isomorphism
• real rank zero
• isometry