Non-Linear New Product $A^*B-B^*A$ 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 $A\diamond B = A^{*}B - B^{*}A$ 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.