Annals of Functional Analysis

Nonlinear maps preserving the Jordan triple -product on von Neumann algebras

Changjing Li, Fangyan Lu, and Ting Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This article investigates a bijective map Φ between two von Neumann algebras, one of which has no central abelian projections, satisfying Φ([[A,B],C])=[[Φ(A),Φ(B)],Φ(C)] for all A,B,C in the domain, where [A,B]=ABBA* is the skew Lie product of A and B. We show that the map Φ(I)Φ is a sum of a linear -isomorphism and a conjugate linear -isomorphism, where Φ(I) is a self-adjoint central element in the range with Φ(I)2=I.

Article information

Source
Ann. Funct. Anal., Volume 7, Number 3 (2016), 496-507.

Dates
Received: 30 October 2015
Accepted: 11 February 2016
First available in Project Euclid: 22 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.afa/1471876886

Digital Object Identifier
doi:10.1215/20088752-3624940

Mathematical Reviews number (MathSciNet)
MR3540447

Zentralblatt MATH identifier
1351.47028

Subjects
Primary: 47B48: Operators on Banach algebras
Secondary: 46L10: General theory of von Neumann algebras

Keywords
Jordan triple $*$-product isomorphism von Neumann algebras

Citation

Li, Changjing; Lu, Fangyan; Wang, Ting. Nonlinear maps preserving the Jordan triple $*$ -product on von Neumann algebras. Ann. Funct. Anal. 7 (2016), no. 3, 496--507. doi:10.1215/20088752-3624940. https://projecteuclid.org/euclid.afa/1471876886


Export citation

References

  • [1] Z. Bai and S. Du, Maps preserving products $XY-YX^{*}$ on von Neumann algebras, J. Math. Anal. Appl. 386 (2012), no. 1, 103–109.
  • [2] M. Brešar and M. Fošner, On rings with involution equipped with some new product, Publ. Math. Debrecen 57 (2000), no. 1–2, 121–134.
  • [3] J. Cui and C.-K. Li, Maps preserving product $XY-YX^{\ast}$ on factor von Neumann algebras, Linear Algebra Appl. 431 (2009), no. 5–7, 833–842.
  • [4] L. Dai and F. Lu, Nonlinear maps preserving Jordan $*$-products, J. Math. Anal. Appl. 409 (2014), no. 1, 180–188.
  • [5] D. Huo, B. Zheng, and H. Liu, Nonlinear maps preserving Jordan triple $\eta$-$*$-products, J. Math. Anal. Appl. 430 (2015), no. 2, 830–844.
  • [6] D. C. Kleinecke, On operator commutators, Proc. Amer. Math. Soc. 8 (1957), 535–536.
  • [7] C. Li, F. Lu, and X. Fang, Nonlinear mappings preserving new product $XY+YX^{*}$ on factor von Neumann algebras, Linear Algebra Appl. 438 (2013), no. 5, 2339–2345.
  • [8] C. R. Miers, Lie homomorphisms of operator algebras, Pacific J. Math. 38 (1971) 717–735.
  • [9] L. Molnár, A condition for a subspace of $\mathcal{B}$(H) to be an ideal, Linear Algebra Appl. 235 (1996), 229–234.
  • [10] P. Šemrl, On Jordan $*$-derivations and an application, Colloq. Math. 59 (1990), no. 2, 241–251.
  • [11] P. Šemrl, Quadratic functionals and Jordan $\ast$-derivations, Studia Math. 97 (1991), no. 3, 157–165.
  • [12] P. Šemrl, Quadratic and quasi-quadratic functionals, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1105–1113.