Jordan $*-$homomorphisms between unital $C^*-$algebras

TL;DR

This paper proves that certain approximate algebraic mappings between unital $C^*$-algebras are actually Jordan homomorphisms, and explores their stability using fixed point methods, extending understanding of algebraic structure preservation.

Contribution

It establishes conditions under which almost unital almost linear maps satisfying specific functional equations are genuine Jordan homomorphisms, including stability results via fixed point techniques.

Findings

Almost unital almost linear maps satisfying the functional equation are Jordan homomorphisms.

For real rank zero $C^*$-algebras, the result holds for continuous maps.

Hyers--Ulam--Rassias stability of Jordan $*$-homomorphisms is demonstrated.

Abstract

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