Nonlinear $\xi$-Jordan *-Triple Derivable Mappings on Factor von Neumann Algebras

Zhang Fangjuan,1, Zhu Xinhong2

 基金资助: 国家自然科学基金.  11601420陕西省自然科学基础研究计划资助项目.  2018JM1053

 Fund supported: the NSFC.  11601420the Natural Science Basic Research Plan in Shaanxi Province.  2018JM1053

Abstract

Let ${\cal A}$ be a factor von Neumann algebra and $\xi$ be a non-zero complex number. A nonlinear map $\phi:\mathcal A\rightarrow\mathcal A$ has been demonstrated to satisfy $\phi(A\diamond_{\xi}B\diamond_{\xi}C)=\phi(A)\diamond_{\xi}B\diamond_{\xi}C+A\diamond_{\xi}\phi(B)\diamond_{\xi}C+A\diamond_{\xi}B\diamond_{\xi}\phi(C)$ for all $A, B, C\in\mathcal A$ if and only if $\phi$ is an additive *-derivation and $\phi(\xi A)=\xi\phi(A)$ for all $A\in\mathcal A.$

Keywords： ξ-Jordan *-triple derivable mapping ; von Neumann algebra ; *-Derivation

Zhang Fangjuan, Zhu Xinhong. Nonlinear $\xi$-Jordan *-Triple Derivable Mappings on Factor von Neumann Algebras. Acta Mathematica Scientia[J], 2021, 41(4): 978-988 doi:

1 引言

${\cal A} $$* -代数, \xi 是非零复数, A, B\in{\cal A}$$ \xi$-Jordan $*$-积定义为$A\diamond_{\xi}B = AB+\xi BA^{*}, $$1 -Jordan * -积通常记为 A\bullet B = AB+BA^{*},$$ -1$-Jordan $*$- 积(斜Lie积)通常记为$[A, B]_{*} = AB-BA^{*}.$近年来, 相关的研究吸引了许多作者的注意(参看文献[1-13]). 文献[1]中P. Šemrl在量子函数中首先引入并研究了$-1$-Jordan $*$-积. 文献[2]研究了von Neumann代数(无中心交换投影)到${\cal B(H)}$上的非线性$\xi$-Jordan $*$-可导映射.

$T = \phi(A_{12}+A_{21})-\phi(A_{12})-\phi(A_{21}). $$I\diamond_{\xi}\frac{P_{1}-\frac{1}{\overline{\xi}}P_{2}}{1+\xi}\diamond_{\xi}A_{12} = 0 和断言1得 所以 I\diamond_{\xi}\frac{P_{1}-\frac{1}{\overline{\xi}}P_{2}}{1+\xi}\diamond_{\xi}T = 0, 由此可得 (1+\xi)T_{11}-(1+\frac{1}{\overline{\xi}})T_{22}+(\xi-\frac{1}{\overline{\xi}})T_{21} = 0.$$ \xi\neq 0, -1 $$T_{11} = T_{22} = 0. 又由 A_{12}\diamond_{\xi}P_{1}\diamond_{\xi}I = 0 所以 T\diamond_{\xi}P_{1}\diamond_{\xi}I = 0, 由此可得 (1+|\xi|^{2})T_{21}+2\xi T_{21}^{*} = 0, 进而 T_{21} = 0. 类似可得 T_{12} = 0. 因此 \phi(A_{12}+A_{21}) = \phi(A_{12})+\phi(A_{21}). 断言4 设 i, j, k\in\{1, 2\},$$ A_{kk}\in {\cal A}_{kk}, A_{ij}\in {\cal A}_{ij}, i\neq j, $$\phi(A_{kk}+A_{ij}) = \phi(A_{kk})+\phi(A_{ij}). 下面只证明 i = k = 1, j = 2, 其他情形同理可得. 令 T = \phi(A_{11}+A_{12})-\phi(A_{11})-\phi(A_{12}).$$ I\diamond_{\xi}\frac{P_{2}}{1+\xi}\diamond_{\xi}A_{11} = 0$和断言1得

$T = \phi(A_{11}+A_{12}+A_{21})-\phi(A_{11})-\phi(A_{12})-\phi(A_{21}).$由断言3得

$A = \sum\limits_{i, i = 1}^{2}A_{ij}, B = \sum\limits_{i, i = 1}^{2}B_{ij}, T = \phi(A+B)-\phi(A)-\phi(B).$任取$X_{12}\in {\cal A}_{12},$由断言5, 断言6和断言7得

$$$X_{12}T+\xi TX_{12}^{*} = 0$$$

$$$\phi(\text{i}{\mathbb R}I)\subseteq\text{i}{\mathbb R}I.$$$

$$$\phi({\mathbb C}I)\subseteq{\mathbb C}I.$$$

$\lambda+\overline{\lambda} = 0, $$\lambda\in\text{i}{\mathbb R}I, 于是存在 \lambda_{1}\in{\mathbb R}, 使得 $$\phi(I) = \text{i}\lambda_{1}I.$$ P_{i}\in{\cal A}, i = 1, 2 为非平凡投影, 由(2.5) 式得 $$\phi(P_{i})^{*} = \phi(P_{i})-2\text{i}\lambda_{1}P_{i}, i = 1, 2.$$ i, j = 1, 2, i\neq j, 由(2.6) 式可得 \begin{eqnarray} 0& = &\phi(P_{i}\bullet P_{j}\bullet P_{i}){}\\ & = &\phi(P_{i})\bullet P_{j}\bullet P_{i}+P_{i}\bullet\phi(P_{j})\bullet P_{i}+P_{i}\bullet P_{j}\bullet \phi(P_{i}){}\\ & = &P_{j}\phi(P_{i})P_{i}+P_{i}\phi(P_{i})P_{j}+2P_{i}\phi(P_{j})P_{i}+\phi(P_{j})P_{i}+P_{i}\phi(P_{j}). \end{eqnarray} (2.7) 式两边同乘 P_{i} $$P_{i}\phi(P_{j})P_{i} = 0.$$ (2.7) 式左乘 P_{i} 右乘 P_{j} $$P_{i}\phi(P_{i})P_{j}+P_{i}\phi(P_{j})P_{j} = 0.$$ (2.7) 式左乘 P_{j} 右乘 P_{i} $$P_{j}\phi(P_{i})P_{i}+P_{j}\phi(P_{j})P_{i} = 0.$$ 对所有 A_{ji}\in{\cal A}_{ji}, 由(2.5) 式和(2.6) 式得 上式左乘 P_{j} 右乘 P_{i}$$ P_{j}\phi(P_{j})A_{ji}+A_{ji}\phi(P_{j})P_{i} = 0.$结合(2.8) 式可得$P_{j}\phi(P_{j})A_{ji} = 0.$由于${\cal A}$是素的, 所以

$$$P_{j}\phi(P_{j})P_{j} = 0.$$$

$$$\phi(I) = \phi(P_{i})+\phi(P_{j}) = P_{i}\phi(P_{i})P_{j}+P_{j}\phi(P_{i})P_{i}+P_{i}\phi(P_{j})P_{j}+P_{j}\phi(P_{j})P_{i} = 0.$$$

$$$\phi(I) = 0.$$$

$$$\phi((1+\xi)A) = (1+\xi)\phi(A), \forall A\in{\cal A}.$$$

$\phi(I\diamond_{\xi}\text{i}I\diamond_{\xi}\text{i}I) = \phi(\text{i}I\diamond_{\xi}\text{i}I\diamond_{\xi}I)$

$$$\phi(\text{i}I)^{*} = -\phi(\text{i}I).$$$

$$$\phi(\text{i}I) = \text{i}\phi(I).$$$

$$$\phi(I)^{*} = \phi(I).$$$

$\begin{eqnarray} \phi(\text{i}(1-2\xi+|\xi|^{2})I) & = &\phi(\text{i}I\diamond_{\xi}I\diamond_{\xi}I){}\\ & = &\phi(\text{i}I)\diamond_{\xi}I\diamond_{\xi}I+\text{i}I\diamond_{\xi}\phi(I)\diamond_{\xi}I+\text{i}I\diamond_{\xi}I\diamond_{\xi}\phi(I){}\\ & = &\phi(\text{i}I)-2\xi\phi(\text{i}I)+|\xi|^{2}\phi(\text{i}I)+2\text{i}\phi(I)-4\text{i}\xi\phi(I)+2\text{i}|\xi|^{2}\phi(I). \end{eqnarray}$

$\begin{eqnarray} \phi(\text{i}(1+2\xi+|\xi|^{2})I)& = &\phi(I\diamond_{\xi}I\diamond_{\xi}\text{i}I){}\\ & = &\phi(I)\diamond_{\xi}I\diamond_{\xi}\text{i}I+I\diamond_{\xi}\phi(I)\diamond_{\xi}\text{i}I+I\diamond_{\xi}I\diamond_{\xi}\phi(\text{i}I){}\\ & = &2\text{i}\phi(I)+4\text{i}\xi\phi(I)+2\text{i}|\xi|^{2}\phi(I)+\phi(\text{i}I)+2\xi\phi(\text{i}I)+|\xi|^{2}\phi(\text{i}I). \end{eqnarray}$

(2.22) 式加(2.23) 式得

$$$\phi(\text{i}(1+|\xi|^{2})I) = \phi(\text{i}I)+2\text{i}\phi(I)+|\xi|^{2}\phi(\text{i}I)+2\text{i}|\xi|^{2}\phi(I).$$$

$\begin{eqnarray} \phi(\text{i}(1-|\xi|^{2})I)& = &\phi(I\diamond_{\xi}\text{i}I\diamond_{\xi}I){}\\ & = &\phi(I)\diamond_{\xi}\text{i}I\diamond_{\xi}I+I\diamond_{\xi}\phi(\text{i}I)\diamond_{\xi}I+I\diamond_{\xi}\text{i}I\diamond_{\xi}\phi(I){}\\ & = &\text{i}\phi(I)-\text{i}|\xi|^{2}\phi(I)+\phi(\text{i}I)-|\xi|^{2}\phi(\text{i}I)+\text{i}\phi(I)-\text{i}|\xi|^{2}\phi(I). \end{eqnarray}$

(2.24) 式加(2.25) 式得$4\text{i}\phi(I) = 0,$

$$$\phi(I) = 0.$$$

$$$\phi(\text{i}I) = 0.$$$

$$$\phi(|\xi|^{2}A) = |\xi|^{2}\phi(A).$$$

$$$\phi(\xi A^{*}) = \xi\phi(A)^{*}.$$$

$$$\phi(\xi A^{*}) = \xi\phi(A^{*}), \forall A\in{\cal A}.$$$

$$$\phi(A^{*}) = \phi(A)^{*}, \forall A\in{\cal A}.$$$

$$$\phi(\text{i}A) = \text{i}\phi(A), \forall A\in{\cal A}.$$$

$\begin{eqnarray} &&\phi(AB+\xi AB+\xi BA^{*}+|\xi|^{2}BA^{*}){}\\ & = &\phi(I\diamond_{\xi}A\diamond_{\xi}B){}\\ & = &I\diamond_{\xi}\phi(A)\diamond_{\xi}B+I\diamond_{\xi}A\diamond_{\xi}\phi(B){}\\ & = &\phi(A)B+\xi\phi(A)B+\xi B\phi(A)^{*}+|\xi|^{2}B\phi(A)^{*}{}\\ &&+A\phi(B)+\xi A\phi(B)+\xi\phi(B)A^{*}+|\xi|^{2}\phi(B)A^{*}. \end{eqnarray}$

$\begin{eqnarray} &&\phi(AB+\xi AB-\xi BA^{*}-|\xi|^{2}BA^{*}){}\\ & = &\phi(I\diamond_{\xi}\text{i}A\diamond_{\xi}(-\text{i}B)){}\\ & = &I\diamond_{\xi}\phi(\text{i}A)\diamond_{\xi}(-\text{i}B)+I\diamond_{\xi}\text{i}A\diamond_{\xi}\phi(-\text{i}B){}\\ & = &\phi(A)B+\xi\phi(A)B-\xi B\phi(A)^{*}-|\xi|^{2}B\phi(A)^{*}{}\\ &&+A\phi(B)+\xi A\phi(B)-\xi\phi(B)A^{*}-|\xi|^{2}\phi(B)A^{*}. \end{eqnarray}$

参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Šemrl P .

Proc Amer Math Soc, 1993, 119 (4): 1105- 1113

Li C J , Lu F Y , Fang X C .

Non-linear $\xi$-Jordan *-derivations on von Neumann algebras

Lin Multi Alg, 2014, 62 (4): 466- 473

Li C J , Zhao F F , Chen Q Y .

Nonlinear skew Lie triple derivations between factors

Acta Math Sin, 2016, 32 (7): 821- 830

Zhao F F , Li C J .

Nonlinear maps preserving the Jordan triple *-product between factors

Indagat Math, 2018, 29 (2): 619- 627

Huo D H , Zheng B D , Liu H Y .

Nonlinear maps preserving Jordan triple $\eta$-*-products

J Math Anal Appl, 2015, 430 (2): 830- 844

Zhang F J .

Nonlinear skew Jordan derivable maps on factor von Neumann algebras

Lin Multi Alg, 2016, 64 (10): 2090- 2103

Yu W Y , Zhang J H .

Nonlinear *-Lie derivations on factor von Neumann algebras

Lin Alg Appl, 2012, 437 (8): 1979- 1991

Zhang F J , Qi X F , Zhang J H .

Nonlinear *-Lie higher derivations on factor von Neumann algebras

B Iran Math Soc, 2016, 42 (3): 659- 678

Zhang F J .

Nonlinear preserving product $XY-\xi YX^{*}$ on prime *-ring

Acta Math Sin, 2014, 57 (4): 775- 784

Taghavi A , Nouri M , Razeghi M , Darvish V .

Non-linear $\lambda$-Jordan triple *-derivation on prime *-algebras

Rochy MT J Math, 2018, 48 (8): 2705- 2716

Zhao F F , Li C J .

Nonlinear *-Jordan triple derivations on von Neumann algebras

Math Slovaca, 2018, 68 (1): 163- 170

von Neumann代数上的Lie可导映射

Yang L C , An R L .

Lie derivable maps on von Neumann algebras

Acta Math Sci, 2018, 38A (5): 864- 872

${\cal J}$-子空间格代数上中心化子和广义导子的刻画

QI X F .

Characterization of centralizers and generalized derivations on ${\cal J}$-subspace lattice algebras

Acta Math Sci, 2014, 34A (2): 463- 472

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