Strong skew commutativity preserving maps on rings with involution

Let R be a unital ^∗-ring with the unit I.Assume that R contains a symmetric idempotent P which satisfies ARP=0 implies A=0 and AR (I-P)=0 implies A=0.In this paper,it is shown that a surjective map Φ:R→R is strong skew commutativity preserving (that is,satisfies Φ(A)Φ(B)-Φ(B)Φ(A)^∗=AB -BA^∗ for all A,B ∈ R) if and only if there exist a map f:R→Z_S (R) and an element Z ∈ Z_S (R) with Z²=I such that Φ(A)=ZA+f (A) for all A ∈ R,where Z_S (R) is the symmetric center of R.As applications,the strong skew commutativity preserving maps on unital prime ∗-rings and von Neumann algebras with no central summands of type I₁ are characterized.