Strong skew commutativity preserving maps on rings with involution

Abstract

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→ZS(R) and an element Z ∈ ZS(R) with Z2 = I such that Φ(A) = ZA + f(A) for all A ∈ R, where ZS(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.