Nonlinear Maps Preserving the Jordan Triple *-Product on Factor von Neumann Algebras

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two factor von Neumann algebras. For $A,B \in \mathcal{A}$, define by [A,B]_* = AB − BA* the skew Lie product of A and B. In this article, it is proved that a bijective map $\Phi :\mathcal{A} \to \mathcal{B}$ satisfies Φ([[A,B]_*,C]_*) = [[Φ(A),Φ(B)]_*,Φ(C)]_* for all $A,B,C \in \mathcal{A}$ if and only if Φ is a linear *-isomorphism, or a conjugate linear *- isomorphism, or the negative of a linear *-isomorphism, or the negative of a conjugate linear *-isomorphism.

Keywords

Jordan triple *-product / Isomorphism / von Neumann algebras

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