Approximately n--Jordan derivations: A fixed point approach

TL;DR

This paper studies the stability of n-Jordan derivations on Banach and C*-algebras using fixed point methods, showing that approximate derivations are close to exact derivations and establishing uniqueness.

Contribution

It introduces a fixed point approach to analyze the stability of n-Jordan derivations and demonstrates the existence and uniqueness of exact derivations near approximate ones.

Findings

Fixed point methods effectively analyze derivation stability.

Approximate *-Jordan derivations are close to true *-derivations.

Uniqueness of *-derivations near approximate derivations in C*-algebras.

Abstract

Let $n\in \Bbb N-\{1\},$ and let $A$ be a Banach algebra. An additive map $D: A\to A$ is called n-Jordan derivation if $$D(a^n)=D(a)a^{n-1}+aD(a)a^{n-2}+...+a^{n-2}D(a)a+a^{n-1}D(a),$$ for all $a \in {A}$. Using fixed point methods, we investigate the stability of n--Jordan derivations (n--Jordan $^*-$derivations) on Banach algebras ($C^*-$algebras). Also we show that to each approximate $^*-$Jordan derivation $f$ in a $C^*-$ algebra there corresponds a unique $^*-$derivation near to $f$.