Jordan *-homomorphisms on $C^*$-algebras

TL;DR

This paper studies Jordan *-homomorphisms on $C^*$-algebras, focusing on their stability properties related to a specific functional inequality and equation, and establishes conditions for their superstability and Hyers-Ulam stability.

Contribution

It introduces new stability results for Jordan *-homomorphisms on $C^*$-algebras linked to a particular functional inequality and equation.

Findings

Proves superstability of Jordan *-homomorphisms.

Establishes generalized Hyers-Ulam stability.

Provides conditions under which approximate solutions are close to true homomorphisms.

Abstract

In this paper, we investigate Jordan *-homomorphisms on $C^*$-algebras associated with the following functional inequality $\|f(\frac{b-a}{3})+f(\frac{a-3c}{3})+f(\frac{3a+3c-b}{3})\| \leq \|f(a)\|.$ We moreover prove the superstability and the generalized Hyers-Ulam stability of Jordan *-homomorphisms on $C^*$-algebras associated with the following functional equation $$f(\frac{b-a}{3})+f(\frac{a-3c}{3})+f(\frac{3a+3c-b}{3})=f(a).$$