Nonlinear mixed Jordan triple *-derivations on factors

TL;DR

This paper characterizes additive *-derivations on factors by a specific nonlinear mixed Jordan triple *-derivation identity, extending understanding of derivation structures in operator algebras.

Contribution

It proves that maps satisfying a nonlinear mixed Jordan triple *-derivation identity are precisely additive *-derivations on factors.

Findings

Characterization of *-derivations via a nonlinear identity

Extension of derivation theory in factors

Identification of additive *-derivations through a new identity

Abstract

Let $\mathcal{A}$ be a factor with dim$\mathcal{A}\geq2$. For $A, B\in\mathcal{A}$, define by $[A, B]_{*}=AB-BA^{\ast}$ and $A\bullet B=AB+BA^{\ast}$ the new products of $A$ and $B$. In this paper, it is proved that a map $\Phi: \mathcal {A}\rightarrow \mathcal {A}$ satisfies $\Phi([A, B]_{*}\bullet C)=[\Phi(A), B]_{*}\bullet C+[A, \Phi(B)]_{*}\bullet C+[A, B]_{*}\bullet \Phi(C)$ for all $A, B,C\in\mathcal {A}$ if and only if $\Phi$ is an additive $*-$derivation.