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ADDITIVITY OF JORDAN MULTIPLICATIVE MAPS ON JORDAN OPERATOR ALGEBRAS

Runling An, 安潤玲, Jinchuan Hou and 侯晉丨丨丨
Taiwanese Journal of Mathematics
Vol. 10, No. 1, Special Issue: Proceedings of the Workshop on Functional Analysis, 2004 (January 2006), pp. 45-64
Stable URL: http://www.jstor.org/stable/43833644
Page Count: 20
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
ADDITIVITY OF JORDAN MULTIPLICATIVE MAPS ON JORDAN OPERATOR ALGEBRAS
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Abstract

Let H be a Hilbert space and N a nest in H. Denote by Sª(H) the Jordan ring of all self-adjoint operators on H and AlgN the nest algebra associated to N. We show that a bijective map Φ : Sª(H) → Sª(H) satisfying (1) Φ(ABA) = Φ(A)Φ(B)Φ(A) for every pair of A, B, or (2) Φ(AB ǀ BA) – Φ(A)Φ(B) + Φ(B)Φ(A) for every pair of A, B, or (3) Φ(½(AB + BA)) = ½ (Φ(A)Φ(B) + Φ(B)Φ(A)) for every pair of A, B must be additive, that is, a Jordan ring isomorphism. We also show that if a bijective map Φ : AlgN → AlgN satisfies the Jordan multiplicativity of the form (2) or (3), then Φ must be a Jordan isomorphism. Moreover, such Jordan multiplicative maps are characterized completely.

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