Madjid Eshaghi Gordji, Norooz Ghobadipour, and Choonkil Park Semnan University, Semnan University, Hanyang University

Abstract : In this paper, we prove the superstability and the generalized Hyers-Ulam stability of Jordan $*$-homomorphisms between unital $C^*$-algebras associated with the following functional equation $$f\left(\frac{-x+y}{3}\right)+f\left(\frac{x-3z}{3}\right)+f\left(\frac{3x-y+3z}{3}\right)=f(x).$$ Moreover, we investigate Jordan $*$-homomorphisms between unital $C^*$-algebras associated with the following functional inequality $$\left\|f\left(\frac{-x+y}{3}\right)+f\left(\frac{x-3z}{3}\right)+f\left(\frac{3x-y+3z}{3}\right)\right\| \leq \|f(x)\|.$$

Keywords : Jordan $*$-homomorphism, $C^*$-algebra, generalized Hyers-Ulam stability, functional equation and inequality