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Jordan triple systems: properties of the generic minimal polynomial


Hermitian positive Jordan triple systems (HPTJS) correspond to bounded symmetric complex domains. We give a survey of the properties of HPJTS and especially of their generic minimal polynomial. As an application, we give a description of the canonical projective realization of the compactification. It turns out that for a natural normalization, the Euclidean volume of a bounded circled homogeneous complex domain is an integer which is equal to the projective degree of the above compactification. The properties of the generic polynomial also lead to the concept of polynomial morphisms of JTS, for which some open problems are stated.

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[1] FARAUT, Jacques, KORANYI, Adam, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal., 88, 64-89, 1990. [2] FARAUT, Jacques, KORANYI, Adam, Analysis on Symmetric Cones, Clarendon Press, Oxford, 1994. [3] LOOS, Ottmar, Bounded symmetric domains and Jordan pairs, Math. Lectures, Univ. of California, Irvine, 1977. [4] ROOS, Guy, Algebres de composition, Systemses triples de Jordan exceptionnels, pp. 1-84, in G. ROOS, J.P. VIGUE, Systemes triple de Jordan et domains symetriques, Travaux en cours, 43, Hermann, Paris, 1992. [5] ROOS, Guy, Volume of bounded symmetric domains and compactification of Jordan triple systems, Conference on Classical and Quantum Geometry of Homogeneous Spaces, Moscow, 1994, to appear in the Proceedings of the Conference.

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