Nonlinear $*$-Jordan-type derivations on alternative $*$-algebras

TL;DR

This paper characterizes nonlinear $*$-Jordan-type derivations on unital alternative $*$-algebras with specific idempotent elements, proving they are equivalent to additive $*$-derivations, with applications to alternative $W^{*}$-algebras.

Contribution

It establishes that nonlinear $*$-Jordan-type derivations are precisely additive $*$-derivations in this algebraic setting, extending understanding of derivation structures.

Findings

Nonlinear $*$-Jordan-type derivations are additive $*$-derivations.

Characterization applies to unital alternative $*$-algebras with specific idempotents.

Results extend to alternative $W^{*}$-algebras.

Abstract

Let $A$ be an unital alternative $*$-algebra. Assume that $A$ contains a nontrivial symmetric idempotent element $e$ which satisfies $xA \cdot e = 0$ implies $x = 0$ and $xA \cdot (1_A - e) = 0$ implies $x = 0$. In this paper, it is shown that $\Phi$ is a nonlinear $*$-Jordan-type derivation on A if and only if $\Phi$ is an additive $*$-derivation. As application, we get a result on alternative $W^{*}$-algebras.