Strong skew commutativity preserving maps on von Neumann algebras

Abstract

Let M be a von Neumann algebra without central summands of type I1. Assume that Φ:M→M is a surjective map. It is shown that Φ is strong skew commutativity preserving (that is, satisfies Φ(A)Φ(B)−Φ(B)Φ(A)∗=AB−BA∗ for all A,B∈M) if and only if there exists some self-adjoint element Z in the center of M with Z2=I such that Φ(A)=ZA for all A∈M. The strong skew commutativity preserving maps on prime involution rings and prime involution algebras are also characterized.