Preserving mixed skew Lie-Jordan-n-product on von Neumann algebra

Abstract This work focuses on characterizing structure-preserving maps between factor von Neumann algebras with respect to a class of mixed skew Lie–Jordan n n -products. Specifically, we consider expressions of the form A₁, A₂_* A₃ Aₙ, A 1, A 2 ∗ ∘ A 3 ∘ ⋯ ∘ A n, where each Aᵢ S A i ∈ S and S S is a factor von Neumann algebra. Here, the skew Lie product is defined by A₁, A₂_*= A₁A₂ - A₂^*A₁ A 1, A 2 ∗ = A 1 A 2 - A 2 ∗ A 1, and the Jordan product by A₁ A₂ = A₁A₂ + A₂A₁ A 1 ∘ A 2 = A 1 A 2 + A 2 A 1. We prove that any map preserving these mixed products is necessarily a linear * ∗ -isomorphism.