Let \({\mathcal {A}}\) be a factor von Neumann algebra with dim\({\mathcal {A}}\ge 2\). We prove that a map \(\phi : {\mathcal {A}}\rightarrow {\mathcal {A}}\) satisfies \(\phi ([A, B]_{\diamond })=[\phi (A), B]_{\diamond }+[A, \phi (B)]_{\diamond }\) for all \(A, B\in {\mathcal {A}}\) if and only if \(\phi \) is an additive \(*\)-derivation, where \([A, B]_{\diamond }=AB^{*}-BA^{*}\).

中文翻译：

---

因子冯·诺依曼代数的非线性双偏李子导数

令\（{\ mathcal {A}} \）是具有暗\（{\ mathcal {A}} \ ge 2 \）的因子冯·诺伊曼代数。我们证明了地图\（\ phi：{\ mathcal {A}} \ rightarrow {\ mathcal {A}} \）满足\（\ phi（[A，B] _ {\ diamond}）= [\ phi（ A），B] _ {\\钻石} + [A，\ phi（B）] _ {\钻石} \）对于所有\（A，B \ {\ mathcal {A}} \）中的所有且仅当\ （\ phi \）是加法\（* \） -派生，其中\（[A，B] _ {\ diamond} = AB ^ {*}-BA ^ {*} \）。