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http://dx.doi.org/10.4134/JKMS.2006.43.3.507

JORDAN AUTOMORPHIC GENERATORS OF EUCLIDEAN JORDAN ALGEBRAS  

Kim, Jung-Hwa (Department of Mathematics Kyungpook National University)
Lim, Yong-Do (Department of Mathematics Kyungpook National University)
Publication Information
Journal of the Korean Mathematical Society / v.43, no.3, 2006 , pp. 507-528 More about this Journal
Abstract
In this paper we show that the Koecher's Jordan automorphic generators of one variable on an irreducible symmetric cone are enough to determine the elements of scalar multiple of the Jordan identity on the attached simple Euclidean Jordan algebra. Its various geometric, Jordan and Lie theoretic interpretations associated to the Cartan-Hadamard metric and Cartan decomposition of the linear automorphisms group of a symmetric cone are given with validity on infinite-dimensional spin factors
Keywords
Euclidean Jordan algebra; symmetric cone; Koecher's theorem; Jordan automorphism; global tubular neighborhood theorem; simultaneous diagonalization; Cartan decomposition; metric and spectral geometric mean; spin factor;
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  • Reference
1 S. Lang, Fundamentals of differential geometry, Graduate Texts in Mathematics, 191. Springer-Verlag, New York, 1999
2 J. D. Lawson and Y. Lim, The geometric mean, matrices, metrics, and more, Amer. Math. Monthly 108 (2001), no. 9, 797-812   DOI   ScienceOn
3 Y. Lim, Applications of geometric means on symmetric cones, Math. Ann. 319 (2001), no. 3, 457-468   DOI
4 Y. Lim, Best approximation in Riemannian geodesic submanifolds of positive definite matrices, Canad. J. Math. 56 (2004), no. 4, 776-793   DOI   ScienceOn
5 Y. Lim, J. Kim, and L. Faybusovich, Simultaneous diagonalization on simple Euclidean Jordan algebras and its applications, Forum Math. 15 (2003), no. 4, 639-644   DOI   ScienceOn
6 K. -H. Neeb, A Cartan-Hadamard theorem for Banach-Finsler manifolds, Geom. Dedicata 95 (2002), 115-156   DOI
7 H. Upmeier, Symmetric Banach manifolds and Jordan C*-algebras, North- Holland Mathematics Studies, 104, North-Holland Publishing Co., Amsterdam, 1985
8 J. D. Lawson and Y. Lim, Means on dyadic symmetric sets and polar decompositions, Abh. Math. Sem. Univ. Hamburg 74 (2004), 135-150   DOI
9 W. Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Dusseldorf-Johannesburg, 1973
10 L. Faybusovich, A Jordan-algebraic approach to potential-reduction algorithms, Math. Z. 239 (2002), no. 1, 117-129   DOI
11 Y. Lim, Geometric means on symmetric cones, Arch. Math. (Basel) 75 (2000), no. 1, 39-45   DOI   ScienceOn
12 W. Kaup, Jordan algebras and holomorphy, Functional analysis, holomorphy, and approximation theory, Lecture Notes in Math., 843, Springer, Berlin, 1981
13 M. Koecher, The Minnesota notes on Jordan algebras and their applications, Edited, annotated and with a preface by Aloys Krieg and Sebastian Walcher, Lecture Notes in Mathematics, 1710. Springer-Verlag, Berlin, 1999
14 L. Faybusovich and T. Tsuchiya, Primal-dual algorithms and infinite-dimen-sional Jordan algebras of finite rank, Math. Program. 97 (2003), no. 3, Ser. B, 471-493   DOI
15 B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ecole Norm. Sup. (4) 6 (1973), 413-455 (1974)
16 O. Kowalski, Generalized symmetric spaces, Lecture Notes in Mathematics, 805. Springer-Verlag, Berlin-New York, 1980
17 M. Fiedler and V. Ptak, A new positive definite geometric mean of two positive definite matrices, Linear Algebra Appl. 251 (1997), 1-20   DOI   ScienceOn
18 W. Bertram, The geometry of Jordan and Lie structures, Lecture Notes in Math- ematics, 1754. Springer-Verlag, Berlin, 2000
19 J. Faraut and A. Koranyi, Analysis on symmetric cones, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford Univer- sity Press, New York, 1994
20 R. A. Hauser, Self-scaled barriers for semidefinite programming, Numerical Anal- ysis Report DAMTP 2000/NA02, Department of Applied Mathematics and The- oretical Physics, Silver Street, Cambridge, England CB3 9EW, March 2000
21 R. A. Hauser and O. Guler, Self-scaled barrier functions on symmetric cones and their classification, Found. Comput. Math. 2 (2002), no. 2, 121-143   DOI   ScienceOn
22 R. A. Hauser and Y. Lim, Self-scaled barriers for irreducible symmetric cones, SIAM J. Optim. 12 (2002), no. 3, 715-723   DOI   ScienceOn
23 A. Kalliterakis, Estimations a l'infini des fonctions de Bessel associees aux repre- sentations d'une algebre de Jordan, J. Lie Theory 11 (2001), no. 2, 273-303