Non-linear *-Jordan derivations on von Neumann algebras

Abstract

Let A be a factor von Neumann algebra and φ be the * -Jordan2010 Mathematics Subject Classification. 46J10 , 47B48.

Nonlinear $\ast$-Jordan-Type Derivations on von Neumann Algebras

“…Note that, unlike von Neumann algebras which are always weakly closed, a standard operator algebra is not necessarily closed. The current work together with [7,[10][11][12][13][23][24][25][26] indicates that it is feasible to investigate * -Jordan-type derivations and * -Lie-type derivations on operator algebras under a unified framework-η- * -Jordan-type derivations. We have good reasons to believe that characterizing η- * -Jordan-type derivations on operator algebras is also of great interest.…”

“…Let H be a complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. Li et al in [10] showed that if [26]. Taghavi et al [23] and Zhang [25] independently investigate * -Jordan derivations on factor von Neumann algebras, respectively. It turns out that each nonlinear * -Jordan derivation on a factor von Neumann algebra is an additive * -derivation.…”

“…η- * -Jordan 2-derivations, η- * -Jordan 3-derivations and η- * -Jordan n-derivations are collectively referred to as η- * -Jordan-type derivations. η- * -Jordan-type derivations on operator algebras are intensively studied by several authors,[7,[10][11][12][13][23][24][25][26] . A basic question in this line is to investigate whether each nonlinear η- * -Jordan-type derivation on an operator algebra A with * is an additive * -derivation.…”

Non-Linear New Product $A^*B-B^*A$ Derivations on $\ast$-Algebras

“…In [15], we showed that * -Jordan derivation map on every factor von Neumann algebra A ⊆ B(H) is additive * -derivation.…”

Non-linear $\ast$-Jordan triple derivation on prime $\ast$-algebras

“…In [9] we showed that * -Jordan derivation map (i.e., φ(A ⋄ B) = φ(A) ⋄ B + A ⋄ φ(B)) on every factor von Neumann algebra A ⊆ B(H) is additive * -derivation. Moreover, in [7] we reduced the assumptions on the map φ.…”