Illinois Journal of Mathematics

Maps preserving zero Jordan products on Hermitian operators

Mikhail A. Chebotar, Wen-Fong Ke, and Pjek-Hwee Lee

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Let $H$ be a separable complex Hilbert space and $B_s(H)$ the Jordan algebra of all Hermitian operators on $H$. Let $\theta:B_s(H)\to B_s(H)$ be a surjective ${\mathbb R}$-linear map which is continuous in the strong operator topology such that $\theta(x)\theta(y)+\theta(y)\theta(x)=0$ for all $x,y\in B_s(H)$ with $xy+yx=0$. We show that $\theta(x)=\lambda uxu^*$ for all $x\in B_s(H)$, where $\lambda$ is a nonzero real number and $u$ is a unitary or anti-unitary operator on $H$.

Article information

Illinois J. Math., Volume 49, Number 2 (2005), 445-452.

First available in Project Euclid: 13 November 2009

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Zentralblatt MATH identifier

Primary: 47B48: Operators on Banach algebras
Secondary: 16W10: Rings with involution; Lie, Jordan and other nonassociative structures [See also 17B60, 17C50, 46Kxx] 17C30: Associated groups, automorphisms 46L70: Nonassociative selfadjoint operator algebras [See also 46H70, 46K70]


Chebotar, Mikhail A.; Ke, Wen-Fong; Lee, Pjek-Hwee. Maps preserving zero Jordan products on Hermitian operators. Illinois J. Math. 49 (2005), no. 2, 445--452. doi:10.1215/ijm/1258138027.

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