Nonlinear Maps Preserving Mixed Product on Factors

Mappings preserving sum of products a◊b + b*a (resp., a*◊b + ab*) on ∗-algebras

“…These products play an important role in some research topics and their studies have recently attracted the attention of some authors (for example, see [1], [4], [5] and [6] and for other products see [2], [3], [7] and [8]). In particular, the authors in [1], [4] and [6] studied bijective mappings preserving the new products mentioned above.…”

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.…”

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

“…Recently, nonlinear maps preserving the products of the mixture of (skew) Lie product and Jordan *product have received a fair amount of attention (see [5,6,8,9,[21][22][23][24]). For example, C. Li et al studied the nonlinear maps preserving the skew Lie triple product [[A, B] * , C] * (see [6,9]) and the Jordan triple * -product A • B • C (see [8,24]) on von Neumann algebras.…”