Non-Linear New Product $A^{\ast }B-B^{\ast }A$ Derivations on $\ast$-Algebras

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 $A\diamond B=A^{\ast }B-B^{\ast }A$ 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.