Dolinar, Gregor; Kuzma, Bojan; Stopar, Nik Orthomaps on formally real simple Jordan algebras. (English) Zbl 1429.17031 Oper. Matrices 13, No. 2, 447-460 (2019). 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 Keywords:formally real Jordan algebra; general preserves of zeros of Jordan product; Jordan maps × Cite Format Result Cite Review PDF 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.