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

Let R be a unital prime -ring containing a nontrivial symmetric idempotent. For A,BR, the skew commutator and 2-skew commutator are defined, respectively, by [A,B]=AB-BA and [A,B]2=[A,[A,B]]. Let phi:RR be a surjective map. We show that (1) phi satisfies [phi(A),phi(B)]=[A,B] for all A,BR if and only if there exists {-1,1} such that phi(A)=A for all AR; (2) phi satisfies [phi(A),phi(B)]2=[A,B]2 for all A,BR if and only if there exists CS with 3=I such that phi(A)=A for all AR, where I is the unit of R and CS is the symmetric extend centroid of R. This is then applied to prime 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.