Nonlinear mixed Jordan triple *-derivations on *-algebras

TL;DR

This paper characterizes nonlinear mixed Jordan triple *-derivations on unital *-algebras, showing they are equivalent to additive *-derivations under certain conditions, with applications to various classes of operator algebras.

Contribution

It establishes a characterization of nonlinear mixed Jordan triple *-derivations as additive *-derivations on unital *-algebras, extending to important classes of operator algebras.

Findings

Such maps are additive *-derivations under mild conditions.

The result applies to prime *-algebras, von Neumann algebras, and standard operator algebras.

Provides a new understanding of the structure of these derivations.

Abstract

Let $\mathcal {A}$ be a unital $\ast$-algebra. 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, under some mild conditions on $\mathcal {A}$, it is shown that a map $\Phi:\mathcal {A}\rightarrow \mathcal {A}$ satisfies $\Phi([A\bullet B, C]_{*})=[\Phi(A)\bullet B, C]_{*}+[A\bullet \Phi(B), C]_{*}+[A\bullet B, \Phi(C)]_{*}$ for all $A, B,C\in\mathcal {A}$ if and only if $\Phi$ is an additive $*-$derivation. In particular, we apply the above result to prime $\ast$-algebras, von Neumann algebras with no central summands of type $I_1$, factor von Neumann algebras and standard operator algebras.