A Note on Nonlinear Mappings Preserving the Bi-Skew Jordan-Type Product on Factor von Neumann Algebras

Let E and F be two factor von Neumann algebras such that E contains a nontrivial symmetric idempotent element e and an identity element I, with dim(E)≥2. In this article, we consider a bijective map ϑ between E and F satisfying ϑ(ν1★ν2★ν3★⋯★νn)=ϑ(ν1)★ϑ(ν2)★ϑ(ν3)★⋯★ϑ(νn) for all νi∈E(i=1,2,…,n), where νi★νj=νi∗νj+νj∗νi is the bi-skew Jordan product of νi, νj for any 1≤i,j≤n, and n≥2 is a fixed positive integer. We prove that ϑ or −ϑ is a conjugate linear ∗-isomorphism or a linear ∗-isomorphism. Moreover, for n=2 and n=3, similar results were obtained by Li and Zhang. In this work, we characterize nonlinear bijective maps preserving the n-product for any n≥2. Thus, our result is more general than both of these earlier results.