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, satisfiesis 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 I 1 are characterized.

Nonlinear maps preserving the mixed product [A ● B,C]* on von Neumann algebras

“…Let A be a * -algebra over the complex field C. For A, B ∈ A, define the skew Lie product of A and B by [A, B] * = AB − BA * and the Jordan * -product of A and B by A • B = AB + BA * . The skew Lie product and the Jordan * -product are fairly meaningful and important in some research topics (see [10][11][12][13][14]25]). They were extensively studied because they naturally arise in the problem of representing quadratic functionals with sesquilinear functionals (see [17][18][19]) and in the problem of characterizing ideals (see [2,16]).…”