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Orthomaps on formally real simple Jordan algebras. (English) Zbl 1429.17031

Summary: We characterize maps on finite-dimensional formally real simple Jordan algebras with the property \(\phi(A \circ B) = \phi(A) \circ \phi(B)\) for all \(A,B\). Although we do not assume additivity it turns out that every such map is either a real linear automorphism or a constant function. The main technique is a reduction to orthomaps, that is, maps which preserve zeros of Jordan product.

MSC:

17C20 Simple, semisimple Jordan algebras
39B52 Functional equations for functions with more general domains and/or ranges
Full Text: DOI

References:

[1] B. R. BAKHADLY, A. E. GUTERMAN, O. V. MARKOVA, Graphs defined by orthogonality, J. Math. Sci. (N.Y.), 207 (2015), no. 5, 698-717. · Zbl 1343.16020
[2] J. L. BRENNER, Matrices of Quaternions, Pacific J. Math. 1, No. 3, (1951), 329-335. · Zbl 0043.01402
[3] K. MCCRIMMON, A Taste of Jordan Algebras, Springer-Verlag, New York, 2004. · Zbl 1044.17001
[4] G. DOLINAR, B. KUZMA, N. STOPAR, The orthogonality relation classifies formally real simple Jordan algebras, submitted. · Zbl 1406.51001
[5] G. DOLINAR, B. KUZMA, N. STOPAR, Characterization of orthomaps on the Cayley plane, Aequationes mathematicae 92 (2018), 243-265. · Zbl 1406.51001
[6] J. FARAUT, A. KORANYI´, Analysis on symmetric cones, Oxford Mathematical Monographs, 1994. · Zbl 0841.43002
[7] C. A. FAURE, An elementary proof of the fundamental theorem of projective geometry, Geom. Dedicata 90, (2002), 145-151. · Zbl 0996.51001
[8] A. FOSNERˇ, B. KUZMA, T. KUZMA, N.-S. SZE, Maps preserving matrix pairs with zero Jordan product, Linear and Multilinear Algebra 59 (2011), 507-529. · Zbl 1222.15031
[9] F. R. HARVEY, Spinors and Calibrations, Academic Press, San Diego, 1990. · Zbl 0694.53002
[10] P. JORDAN, J.VONNEUMANN, E. WIGNER, On an algebraic generalization of the quantum mechanical formalism, Ann. Math., 36 (1934), 29-64. · JFM 60.0902.02
[11] P. JORDAN, ¨Uber Verallgemeinerungsm¨oglichkeiten des Formalismus der Quantenmechanik, Nachr. Ges. Wiss. G¨ottingen (1933), 209-217. · JFM 59.0796.02
[12] L. RODMAN, Topics in quaternion linear algebra, Princeton University Press, New-Jersey, 2014. · Zbl 1304.15004
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