von Neumann代数上的混合Lie可乘映射

Mixed Lie Multiplicative Maps on von Neumann Algebras

摘要: 设$ℳ$ 和$\mathcal{N}$ 是无$I_1$ 或$I_2$ 型中心直和项的von Neumann代数，其单位元分别为$I$ 和$I^{\prime }$ 。本文证明非线性双射$\Phi :ℳ\to \mathcal{N}$ 混合Lie可乘，即$\Phi \left({\left[\left[A,B\right],C\right]}_{\ast }\right)={\left[\left[\Phi \left(A\right),\Phi \left(B\right)\right],\Phi \left(C\right)\right]}_{\ast },\forall A,B,C\in ℳ$ ，当且仅当存在线性*-同构和共轭线性*-同构的直和$\Psi :ℳ\to \mathcal{N}$ 使得$\Phi \left(A\right)=\Phi \left(I\right)\Psi \left(A\right),\forall A\in ℳ$ ，其中$\Phi \left(I\right)\in \mathcal{N}$ 是可逆中心元且$\Phi {\left(I\right)}^2=I^{\prime }$ 。该结论将因子von Neumann代数上的非线性混合Lie可乘双射的结果推广到无$I_1$ 或$I_2$ 型中心直和项的von Neumann代数。

Abstract: Let $ℳ$ and $\mathcal{N}$ be von Neumann algebras with no central summands of type $I_1$ or $I_2$ , $I$ and $I^{\prime }$ be the identities of them. This paper proves that a bijective map $\Phi :ℳ\to \mathcal{N}$ is mixed Lie multiplicative, that is, $\Phi \left({\left[\left[A,B\right],C\right]}_{\ast }\right)={\left[\left[\Phi \left(A\right),\Phi \left(B\right)\right],\Phi \left(C\right)\right]}_{\ast },\forall A,B,C\in ℳ$ if and only if $\Phi \left(A\right)=\Phi \left(I\right)\Psi \left(A\right)$ for all $A\in ℳ$ , where $\Psi :ℳ\to \mathcal{N}$ is a direct sum of a linear *-isomorphism and a conjugate linear *-isomorphism, $\Phi \left(I\right)$ is a central element in $\mathcal{N}$ with $\Phi {\left(I\right)}^2=I^{\prime }$ . The results about mixed Lie multiplicative maps on factor von Neumann algebras are generalized to von Neumann algebras with no central summands of type $I_1$ or $I_2$ .