Strong 2-skew Commutativity Preserving Maps on Prime Rings with Involution

Abstract

Let ${\mathcal {R}}$ be a unital prime $*$-ring containing a nontrivial symmetric idempotent. For $A,B\in {\mathcal {R}}$, the skew commutator and 2-skew commutator are defined, respectively, by ${}_*[A,B]=AB-BA^*$ and ${}_*[A,B]_2= {{}_*[A, {{}_*[A,B]}]}$. Let $\Phi :{\mathcal {R}} \rightarrow {\mathcal {R}}$ be a surjective map. We show that (1) $\Phi$ satisfies ${}_*[\Phi (A),\Phi (B)] = {{}_*[A,B] }$ for all $A, B\in {\mathcal {R}}$ if and only if there exists $\lambda \in \{-1,1\}$ such that $\Phi (A)=\lambda A$ for all $A\in {\mathcal {R}}$; (2) $\Phi$ satisfies ${}_*[\Phi (A),\Phi (B)]_2= {{}_*[A,B]_2}$ for all $A, B\in {\mathcal {R}}$ if and only if there exists $\lambda \in {\mathcal {C}}_S$ with $\lambda ^{3} = I$ such that $\Phi (A) = \lambda A$ for all $A \in {\mathcal {R}}$, where I is the unit of ${\mathcal {R}}$ and ${\mathcal {C}}_S$ is the symmetric extend centroid of ${\mathcal {R}}$. This is then applied to prime $\hbox {C}^*$-algebras, factor von Neumann algebras and indefinite self-adjoint standard operator algebras to get a complete invariant for the identity map and to symmetric standard operator algebras as well as matrix algebras.