Strong skew commutativity preserving maps on von Neumann algebras

Abstract

Let ${\mathcal M}$ be a von Neumann algebra without central summands of type $I_1$. Assume that $\Phi:{\mathcal M}\rightarrow {\mathcal M}$ is a surjective map. It is shown that $\Phi$ is strong skew commutativity preserving (that is, satisfies $\Phi(A)\Phi(B)-\Phi(B)\Phi(A)^*=AB-BA^*$ for all $A,B\in{\mathcal M}$) if and only if there exists some self-adjoint element $Z$ in the center of ${\mathcal M}$ with $Z^2=I$ such that $\Phi(A)=ZA$ for all $A\in{\mathcal M}$. The strong skew commutativity preserving maps on prime involution rings and prime involution algebras are also characterized.Comment: 16 page

Strong (skew) ξ-Lie commutativity preserving maps on algebras

“…However, the inverse is not true generally. In Cui and Park (2012), they proved that, if  is a factor von Neumann algebra, then every strong skew commutativity preserving map Φ on  has the form Φ Qi and Hou (2013) generalized the above result to von Neumann algebras without central summand of type I 1 .…”

Strong Skew Commutativity Preserving Maps on rings

“…In [6], Cui and Park characterized nonlinear surjective strong skew commutativity preserving maps φ on factor von Neumann algebras A, that is, φ has the form φ(A) = ψ(A) + h(A)I for all A ∈ A, where ψ : A → A is a linear bijective Supported map satisfying ψ(A)ψ(B) − ψ(B)ϕ(A) * = AB − BA * for all A, B ∈ A and h is a real functional on A with h(0) = 0. In [11], Qi and Hou generalized Cui and Park's results to prime rings with involution and proved that every nonlinear surjective strong skew commutativity preserving map φ on a unital prime ring A with involution has the form φ(A) = λA + f (A) for all A ∈ A, where λ ∈ {−1, 1} and f is a map from A into Z s (A) (the symmetric center of A). Qi and Hou in [11] also characterized the nonlinear surjective strong skew commutativity preserving maps of von Neumann algebras without central summands of type I 1 .…”

“…In [11], Qi and Hou generalized Cui and Park's results to prime rings with involution and proved that every nonlinear surjective strong skew commutativity preserving map φ on a unital prime ring A with involution has the form φ(A) = λA + f (A) for all A ∈ A, where λ ∈ {−1, 1} and f is a map from A into Z s (A) (the symmetric center of A). Qi and Hou in [11] also characterized the nonlinear surjective strong skew commutativity preserving maps of von Neumann algebras without central summands of type I 1 . In this article, we continue this line of investigation and characterize nonlinear strong skew commutativity preserving maps on general rings with involution.…”

“…Then we have φ(A) = ZA + f (A), completing the proof.Because a unital prime ring with involution satisfies the hypotheses of Theorem 2.1 if it contains a nontrivial idempotent, the following result is immediate from Theorem 2.1, which was obtained in[11, Theorem 2.1].…”