Taiwanese Journal of Mathematics

ADDITIVITY OF JORDAN MULTIPLICATIVE MAPS ON JORDAN OPERATOR ALGEBRAS

Runling An and Jinchuan Hou

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Abstract

Let $H$ be a Hilbert space and $\mathcal{N}$ a nest in $H$. Denote by $S^a(H)$ the Jordan ring of all self-adjoint operators on $H$ and $\mathrm{Alg}\mathcal{N}$ the nest algebra associated to $\mathcal{N}$. We show that a bijective map $\Phi : S^a(H) \to S^a(H)$ satisfying (1) $\Phi(ABA) = \Phi(A) \Phi(B) \Phi(A)$ for every pair of $A,B$, or (2) $\Phi(AB + BA) = \Phi(A) \Phi(B) + \Phi(B) \Phi(A)$ for every pair of $A,B$, or (3) $\Phi(\frac{1}{2} (AB + BA)) = \frac{1}{2} (\Phi(A) \Phi(B) + \Phi(B) \Phi(A))$ for every pair of $A,B$ must be additive, that is, a Jordan ring isomorphism. We also show that if a bijective map $\Phi : \mathrm{Alg}\mathcal{N} \to \mathrm{Alg}\mathcal{N}$ satisfies the Jordan multiplicativity of the form (2) or (3), then $\Phi$ must be a Jordan isomorphism. Moreover, such Jordan multiplicative maps are characterized completely.

Article information

Source
Taiwanese J. Math., Volume 10, Number 1 (2006), 45-64.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500403798

Digital Object Identifier
doi:10.11650/twjm/1500403798

Mathematical Reviews number (MathSciNet)
MR2186161

Zentralblatt MATH identifier
1107.46047

Subjects
Primary: 46C20: Spaces with indefinite inner product (Krein spaces, Pontryagin spaces, etc.) [See also 47B50] 47B49: Transformers, preservers (operators on spaces of operators)

Keywords
Hilbert spaces automorphisms Jordan product

Citation

An, Runling; Hou, Jinchuan. ADDITIVITY OF JORDAN MULTIPLICATIVE MAPS ON JORDAN OPERATOR ALGEBRAS. Taiwanese J. Math. 10 (2006), no. 1, 45--64. doi:10.11650/twjm/1500403798. https://projecteuclid.org/euclid.twjm/1500403798


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