Journal of the
Korean Mathematical Society JKMS

It is shown that every almost linear mapping $h : \mathcal A \rightarrow \mathcal B$ of a unital $C^*$-algebra $\mathcal A$ to a unital $C^*$-algebra $\mathcal B$ is a homomorphism under some condition on multiplication, and that every almost linear continuous mapping $h : \mathcal A \rightarrow \mathcal B$ of a unital $C^*$-algebra $\mathcal A$ of real rank zero to a unital $C^*$-algebra $\mathcal B$ is a homomorphism under some condition on multiplication. Furthermore, we are going to prove the generalized Hyers-Ulam-Rassias stability of $*$-homomorphisms between unital $C^*$-algebras, and of $\Bbb C$-linear $*$-derivations on unital $C^*$-algebras.

Keywords: homomorphism, $C^*$-algebra of real rank zero, linear derivation, stability

MSC numbers: Primary 47B48, 39B52, 46L05