Let \({\mathcal {A}}\) and \({\mathcal {B}}\) be two factors with dim\({\mathcal {A}}>4\). In this article, it is proved that a bijective map \(\Phi : {\mathcal {A}}\rightarrow {\mathcal {B}}\) satisfies \(\Phi ([A\bullet B, C])=[\Phi (A)\bullet \Phi (B), \Phi (C)]\) for all \(A, B, C\in {\mathcal {A}}\) if and only if \(\Phi \) is a linear \(*\)-isomorphism, or a conjugate linear \(*\)-isomorphism, or the negative of a linear \(*\)-isomorphism, or the negative of a conjugate linear \(*\)-isomorphism.

中文翻译：

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保留因子混合乘积的非线性映射

令\（{\ mathcal {A}} \）和\（{\ mathcal {B}} \）是昏暗\（{\ mathcal {A}}> 4 \）的两个因子。在本文中，证明了双射图\（\ Phi：{\ mathcal {A}} \ rightarrow {\ mathcal {B}} \）满足\（\ Phi（[A \ bullet B，C]）= [\披（A）\子弹\披（B），\披（C）] \）对于所有\（A，B，C \在{\ mathcal {A}} \）当且仅当\（\披\）是线性\（* \）-同构，或者共轭线性\（* \）-同构，或者是线性\（* \）-同构，或者是共轭线性\（* \ ）-同构。