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

Abstract

Let H be a complex Hilbert space, B(H) be the algebra of all bounded linear operators on H and A ⊆ B(H) be a von Neumann algebra without central summands of type I 1 . For arbitrary elements A, B ∈ A, one can define their * -Jordan product in the sense of A ⋄ B = AB + BA * . Let pn(x 1 , x 2 , • • • , xn) be the polynomial defined by n indeterminates x 1 , • • • , xn and their * -Jordan products. In this article, it is shown that a mapping δ : A −→ B(H) satisfies the conditionand only if δ is an additive * -derivation.

Nonlinear Lie-Type Derivations of finitary Incidence Algebras and Related Topics

“…This notion makes the best use of the definition of * -Jordan-type derivation and that of * -Lie-type derivation, see [17,21,[31][32][33]. By the definition, it is clear that every * -Jordan derivation is an * -Jordan 2-derivation and each * -Jordan triple derivation is an * -Jordan 3-derivation.…”

“…Therefore, studying Lie triple derivations enables us to treat both important class of Jordan derivations and Lie derivations simultaneously. More recently, Jordan-type derivations on triangular algebras, prime rings, matrix algebras, nest algebras and von Neumann algebras are considered by Lin, Qi and Zhao et al, see [33,53,56]. An R-bilinear mapping ϕ : A × A −→ A is a Jordan biderivation if it is a Jordan derivation with respect to both components, implying that ϕ(x • y, z) = ϕ(x, z) • y + x • ϕ(y, z) and ϕ(x, y • z) = ϕ(x, y) • z + y • ϕ(x, z) for all x, y ∈ A.…”

“…Those * -Lie-type derivations without the additivity assumption are called nonlinear * -Lie-type derivations. (Non-)Linear * -Lie-type derivations on various algebras are intensively studied by some researchers, see[31][32][33].Question 5.1. Let R be a 2-torsionfree and (n−1)-torsionfree commutative ring with identity, and F I(X, R) be the finitary incidence algebra over a pre-ordered set (X, ≤).…”