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Preservation of Order and Orthogonality on Preduals of Jordan Algebras

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Abstract

We study linear maps between preduals of JBW-algebras that preserve orthogonal decompositions of functionals. We generalize and strengthen results of Bunce and Wright (Pac J Math 158(2):265–272, 1993) obtained for von Neumann algebras by characterizing continuous orthogonally decomposable linear maps between preduals of JBW-algebras. In particular, we show that Banach space adjoint of such morphism \(\Phi \) is typically of the form \(\Phi ^*(y)= c\pi (y)\), where c is a central positive element and \(\pi \) is a Jordan homomorphism. Further we show that the order topology on preduals of JBW-algebras equals to the norm topology. This allows to relax continuity in characterization of orthogonally decomposable isomorphisms and show that order and orthogonality relation in preduals are complete invariants of JBW-algebras.

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Acknowledgements

The authors would like to express their gratitude to the referee for the careful reading of the first version of the manuscript and for the many valuable comments that thoroughly enhanced the presentation of this paper.

Funding

The work of Jan Hamhalter was supported by the project GA23-04776 S " Interplay of algebraic, metric, geometric and topological structures on Banach spaces" and by the project SGS23/056/OHK3/1T/13.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Malta, Msida, Malta

    E. Chetcuti

  2. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technicka 2, 166 27, Prague 6, Czech Republic

    J. Hamhalter

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  1. E. Chetcuti
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  2. J. Hamhalter
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Correspondence to E. Chetcuti.

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Chetcuti, E., Hamhalter, J. Preservation of Order and Orthogonality on Preduals of Jordan Algebras. Results Math 79, 114 (2024). https://doi.org/10.1007/s00025-024-02138-y

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