Nonlinear Maps Preserving the Mixed Product [A ○ B, C]* on Von Neumann Algebras

作者：Li, Changjing^*; Zhao, Yuanyuan; Zhao, Fangfang

来源：Filomat, 2021, 35(8): 2775-2781.

DOI：10.2298/FIL2108775L

• Jordan *-product
• isomorphism
• von Neumann algebras

Let A and B be two von Neumann algebras. For A, B is an element of A, define by [A, B](*) = AB - BA* and A circle B = AB + BA* the new products of A and B. Suppose that a bijective map Phi : A -> B satisfies Phi([A circle B, C](*)) = [Phi(A) circle Phi(B), Phi(C)](*) for all A, B, C is an element of A. In this paper, it is proved that if A and B be two von Neumann algebras with no central abelian projections, then the map Phi(I)Phi is a sum of a linear *-isomorphism and a conjugate linear *-isomorphism, where Phi(I) is a self-adjoint central element in B with Phi(I)(2) = I. If A and B are two factor von Neumann algebras, then Phi is a linear *-isomorphism, or a conjugate linear *-isomorphism, or the negative of a linear *-isomorphism, or the negative of a conjugate linear *-isomorphism.

• 单位
  山东师范大学