Nonlinear maps preserving product \(X^{*}Y+Y^{*}X\) on von Neumann algebras. (English) Zbl 07040690
Summary: Let \(\mathcal {A}\) and \(\mathcal {B}\) be two von Neumann algebras with no central abelian projections. In this paper, it is proved that if a not necessarily linear bijective map \(\Phi :\mathcal {A}\rightarrow \mathcal {B}\) satisfies \(\Phi (A^{*}B+B^{*}A)=\Phi (A)^{*}\Phi (B)+\Phi (B)^{*}\Phi (A)\) for all \(A, B\in \mathcal {A}\), then \(\Phi \) is a sum of a linear \(*\)-isomorphism and a conjugate linear \(*\)-isomorphism.
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