Nonlinear *-Jordan-type derivations on *-algebras

Abstract

Let $\mathcal{𝒜}$ be a unital $\ast$-algebra with the unit $I$. Assume that $\mathcal{𝒜}$ contains a nontrivial projection $P$ which satisfies $X\mathcal{𝒜}P=0$ implies $X=0$ and $X\mathcal{𝒜}\left(I-P\right)=0$ implies $X=0$. In this paper, it is shown that $\mathrm{\Phi }$ is a nonlinear $\ast$-Jordan-type derivation on $\mathcal{𝒜}$ if and only if $\mathrm{\Phi }$ is an additive $\ast$-derivation. As applications, the nonlinear $\ast$-Jordan-type derivations on prime $\ast$-algebras, von Neumann algebras with no central summands of type $I_1$, factor von Neumann algebras and standard operator algebras are characterized.