# Non-Linear New Product $A^*B-B^*A$ Derivations on $\ast$-Algebras

## TL;DR

This paper characterizes additive *-derivations on prime *-algebras via a nonlinear product condition involving the algebra's involution, extending understanding of derivation structures.

## Contribution

It introduces a nonlinear product condition to identify when a map on a prime *-algebra is an additive *-derivation, providing new characterization criteria.

## Key findings

- erivations are characterized by the nonlinear product condition.
- erivations are additive under the given conditions.
- The map tisfies *-derivation properties if tisfies the specified self-adjoint condition.

## Abstract

Let $\mathcal{A}$ be a prime $\ast$-algebra. In this paper, we suppose that $\Phi:\mathcal{A}\to\mathcal{A}$ satisfies $$\Phi(A\diamond B)=\Phi(A)\diamond B+A\diamond\Phi(B)$$ where $A\diamond B = A^{*}B - B^{*}A$ for all $A,B\in\mathcal{A}$ .We will show that if $\Phi(\alpha \frac{I}{2})$ is self-adjoint for $\alpha\in\{1,i\}$ then $\Phi$ is additive $\ast$-derivation.