Nearly Jordan ∗ -Homomorphisms between Unital C ∗ -Algebras

Abstract

Let $A$, $B$ be two unital $C^{*}$-algebras. We prove that every almost unital almost linear mapping $h$ : $A\to B$ which satisfies $h(3^{n}uy+3^{n}yu)=h\left(3^{n}u\right)h\left(y\right)+h\left(y\right)h\left(3^{n}u\right)$ for all $u\in U\left(A\right)$, all $y\in A$, and all $n=0,1,2,\dots$, is a Jordan homomorphism. Also, for a unital $C^{*}$-algebra $A$ of real rank zero, every almost unital almost linear continuous mapping $h:A\to B$ is a Jordan homomorphism when $h(3^{n}uy+3^{n}yu)=h\left(3^{n}u\right)h\left(y\right)+h\left(y\right)h\left(3^{n}u\right)$ holds for all $u\in I_1$ ($A_{\text{sa}}$), all $y\in A$, and all $n=0,1,2,\dots$. Furthermore, we investigate the Hyers- Ulam-Aoki-Rassias stability of Jordan $*$-homomorphisms between unital $C^{*}$-algebras by using the fixed points methods.