Let \({\mathcal {M}}\) be a factor von Neumann algebra on a complex separable Hilbert space \({\mathcal {H}}\) with \(\dim {\mathcal {M}}>1\). We proved that if \(\varPhi :{\mathcal {M}}\rightarrow {\mathcal {M}}\) is a second nonlinear mixed Jordan triple derivable mapping, that is,

$$\begin{aligned} \varPhi (A\circ B\bullet C)=\varPhi (A)\circ B\bullet C+A\circ \varPhi (B)\bullet C+A\circ B\bullet \varPhi (C) \end{aligned}$$

for all \(A,B,C\in {\mathcal {M}}\), then \(\varPhi \) is an additive \(*\)-derivation.

中文翻译：

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因子冯·诺依曼代数上的第二个非线性混合约旦三阶导数映射

令\（{\ mathcal {M}} \）为复式可分离Hilbert空间\（{\ mathcal {H}} \）上\（\ dim {\ mathcal {M}}> 1 \ ）。我们证明，如果\（\ varPhi：{\ mathcal {M}} \ rightarrow {\ mathcal {M}} \）是第二个非线性混合Jordan三次可导映射，即，

$$ \ begin {aligned} \ varPhi（A \ circ B \ bullet C）= \ varPhi（A）\ circ B \ bullet C + A \ circ \ varPhi（B）\ bullet C + A \ circ B \ bullet \ varPhi（C）\ end {aligned} $$

对于{\ mathcal {M}} \中的所有\（A，B，C \），则\（\ varPhi \）是加法\（* \）派生。