Nonlinear mixed Jordan triple *-derivations on *-algebras (2112.04103v1)

Abstract: Let $\mathcal {A}$ be a unital $\ast$-algebra. For $A, B\in\mathcal{A}$, define by $[A, B]{*}=AB-BA^{\ast}$ and $A\bullet B=AB+BA^{\ast}$ the new products of $A$ and $B$. In this paper, under some mild conditions on $\mathcal {A}$, it is shown that a map $\Phi:\mathcal {A}\rightarrow \mathcal {A}$ satisfies $\Phi([A\bullet B, C]{})=[\Phi(A)\bullet B, C]_{}+[A\bullet \Phi(B), C]{*}+[A\bullet B, \Phi(C)]{}$ for all $A, B,C\in\mathcal {A}$ if and only if $\Phi$ is an additive $-$derivation. In particular, we apply the above result to prime $\ast$-algebras, von Neumann algebras with no central summands of type $I_1$, factor von Neumann algebras and standard operator algebras.