## Annals of Functional Analysis

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

#### Abstract

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

#### Article information

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

Dates
Accepted: 11 February 2016
First available in Project Euclid: 22 August 2016

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

#### 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

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