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

Abstract

Let A and B be two von Neumann algebras. For A,B ? A, define by [A,B]* = AB-BA* and A ? B = AB + BA* the new products of A and B. Suppose that a bijective map ? : A ? B satisfies ?([A ? B,C]*) = [?(A)? ?(B),?(C)]* for all A,B,C ? A. In this paper, it is proved that if A and B be two von Neumann algebras with no central abelian projections, then the map ?(I)? is a sum of a linear *-isomorphism and a conjugate linear +-isomorphism, where ?(I) is a self-adjoint central element in B with ?(I)2 = I. If A and B are two factor von Neumann algebras, then ? is a linear *-isomorphism, or a conjugate linear *-isomorphism, or the negative of a linear *-isomorphism, or the negative of a conjugate linear *-isomorphism.

Nonlinear mixed Jordan triple ∗-derivations on factor von Neumann algebras

“…Recently, many authors have studied the isomorphisms and derivations corresponding to the new products of the mixture of (skew) Lie product and Jordan * -product (see [17,26,27,30,33,34]). Z. Yang and J. Zhang [26,27] studied the nonlinear maps preserving the mixed skew Lie triple product [[A, B] * , C] and [[A, B], C] * on factor von Neumann algebras, where [A, B] = AB − BA is the usual Lie product of A and B. Y. Zhou, Z. Yang and J. Zhang [34] studied the structure of the nonlinear mixed Lie triple derivations on prime * -algebras.…”

“…More generally, we say that Φ is a nonlinear Jordan * -derivation or skew Lie derivation if] * for all A, B ∈ A. Many authors have paid more attentions on the problem about Jordan * -derivations, skew Lie derivations and triple derivations, such as Jordan triple * -derivations and skew Lie triple derivations (see [6, 14, 15, 18-20, 25, 28, 29, 31, 32]).Recently, many authors have studied the isomorphisms and derivations corresponding to the new products of the mixture of (skew) Lie product and Jordan * -product (see [17,26,27,30,33,34]). Z. Yang and J. Zhang [26,27] studied the nonlinear maps preserving the mixed skew Lie triple product [[A, B] * , C] and [[A, B], C] * on factor von Neumann algebras, where [A, B] = AB − BA is the usual Lie product of A and B. Y. Zhou, Z. Yang and J. Zhang [34] studied the structure of the nonlinear mixed Lie triple derivations on prime * -algebras.…”