JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (6): 106-111.doi: 10.6040/j.issn.1671-9352.0.2018.257

Previous Articles    

Nonlinear maps preserving mixed Lie triple ξ-product on factor von Neumann algebras

ZHOU You1, ZHANG Jian-hua2*   

  1. 1. School of Mathematics Science, Qufu Normal University, Qufu 273165, Shandong, China;
    2. School of Mathematics and Information Science, Shaanxi Normal University, Xian 710062, Shaanxi, China
  • Published:2019-06-05

Abstract: In this paper, we prove that every bijective map preserving mixed Lie triple ξ-products with ξ≠1 from a factor von Neumann algebra M with dim M >1 into another factor von Neumann algebra N with dim N >1 is of the form A→εΨ(A), where ε∈{1,-1} and Ψ:M→N is a linear or conjugate linear *-isomorphism when ξ∈R and Ψ is a linear *-isomorphism when ξ∈C\R.

Key words: mixed Lie triple ξ-product, factor von Neumann algebras, preserver

CLC Number: 

  • O177.1
[1] BRESAR M, MIERS C R. Commutativity preserving mappings of von Neumann algebras[J]. Canadian Journal of Mathematics, 1993, 45(4):695-708.
[2] JANKO M. A note on Lie product preserving maps on Mn(R)[J]. Mathematica Slovaca, 2016, 66(3):715-720.
[3] YU Xiuping, LU Fangyan. Maps preserving Lie product on B(X)[J]. Taiwanese Journal Mathematics, 2008, 12(3):793-806.
[4] 张芳娟.素*-环上非线性保XY-ξYX* 积[J].数学学报(中文版),2014, 57(4):775-784. ZHANG Fangjuan. Nonlinear preserving product XY-ξYX on prime *-ring[J]. Acta Mathematica Sinica(Chinese Series), 2014, 57(4):775-784.
[5] ZHANG Jianhua, ZHANG Fangjuan. Nonlinear maps preserving Lie products on factor von Neumann algebras[J]. Linear Algebra and its Applications, 2008, 429(1):18-30.
[6] BAI Zhaofang, DU Shuanping. Maps preserving products XY-YX* on von Neumann algebras[J]. Journal of Mathematical Analysis and Applications, 2012, 386(1):103-109.
[7] CUI Jianlian, CHOONKIL P. Maps preserving strong skew lie product on factor von Neumann algebras[J]. Acta Mathematica Scientia, 2012, 32(2):531-538.
[8] CUI Jianlian, LI Chi-Kwong. Maps preserving product XY-YX* on factor von Neumann algebras[J]. Linear Algebra and its Applications, 2009, 431(5-7):833-842.
[9] QI Xiaofei, HOU Jinchuan. Strong skew commutativity preserving maps on von Neumann algebras[J]. Journal of Mathematical Analysis and Applications, 2013, 397(1):362-370.
[10] LIU Lei. Lie triple derivations on factor von Neumann algebras[J]. Bulletin of the Korean Mathematical Society, 2015, 52(2):581-591.
[11] MIERS C R. Lie triple derivations of von Neumann algebras[J]. Proceedings of the American Mathematical Society, 1978, 71(1):57-61.
[12] HUO Donghua, ZHENG Baodong, LIU Hongyu. Nonlinear maps preserving Jordan triple η-*-products[J]. Journal of Mathematical Analysis Applications, 2015, 430(2):830-844.
[13] LI Changjing, LU Fangyan. Nonlinear maps preserving the Jordan triple 1-*-product on von Neumann algebras[J]. Complex Analysis Operator Theory, 2017, 11(1):109-117.
[14] LI Changjing, LU Fangyan, WANG Ting. Nonlinear maps preserving Jordan triple *-product on von Neumann algebras[J]. Annals of Functional Analysis, 2016, 7(3):496-507.
[1] YIN Jun-cheng1,2, CAO Huai-xin1. Wavelet preservers on the Hilbert space L2(R) [J]. J4, 2012, 47(4): 57-61.
[2] FANG Li1, BAI Wei-zu2. Maps preserving the idempotency of products or triple Jordan products of idempotent operators [J]. J4, 2010, 45(12): 98-105.
[3] CUI Yun-Li, ZHANG Jian-Hua. Linear maps preserving zeros of a polynominal on  factor von Neumann algebras [J]. J4, 2009, 44(10): 48-50.
[4] WU Xiaogui, ZHANG Jianhua. Linear maps preserving Jacobi identity on the full matrix algebras [J]. J4, 2009, 44(1): 49-52 .
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
No Suggested Reading articles found!