Nonlinear mappings preserving Jordan-type η-*-products

Abstract

Let η be a nonzero complex number. Let 𝒜 and ℬ be two von Neumann algebras, one of which has no central abelian projections, and let Φ:𝒜→ℬ be not necessarily a linear bijection. For arbitrary elements A,B∈𝒜, one can define their Jordan η-∗-product in the sense of A◇ηB=AB+ηBA∗. Let pn(X1,X2,…,Xn) be the polynomial defined by n indeterminates X1,…,Xn and their Jordan multiple η-∗-product. In this article, it is shown that Φ satisfies the condition Φ(pn(A1,A2,…,An))=pn(Φ(A1),Φ(A2),…,Φ(An)) for all A1,A2,…,An∈𝒜 if and only if one of the following statements holds true: (1)η∈ℝ and there exists a central projection E∈𝒜 such that Φ(E) is a central projection in ℬ, Φ|𝒜E:𝒜E→ℬΦ(E) is a linear ∗-isomorphism and Φ|𝒜(I−E):𝒜(I−E)→ℬ(I−Φ(E)) is a conjugate linear ∗-isomorphism, (2)η∉ℝ and Φ is a linear ∗-isomorphism.