A structure theory for Jordan H * -pairs

A. J. Calderón Martín; C. Martín González

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 1, page 61-77
  • ISSN: 0392-4041

Abstract

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Jordan H * -pairs appear, in a natural way, in the study of Lie H * -triple systems ([3]). Indeed, it is shown in [4, Th. 3.1] that the problem of the classification of Lie H * -triple systems is reduced to prove the existence of certain L * -algebra envelopes, and it is also shown in [3] that we can associate topologically simple nonquadratic Jordan H * -pairs to a wide class of Lie H * -triple systems and then the above envelopes can be obtained from a suitable classification, in terms of associative H * -pairs, of these pairs. In this paper we give a classification theorem for topologically simple non-quadratic Jordan H * -pairs in terms of associative H * -pairs and certain of their anti-isomorphisms. Some consequences of this classification are also stated.

How to cite

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Calderón Martín, A. J., and González, C. Martín. "A structure theory for Jordan $H^*$-pairs." Bollettino dell'Unione Matematica Italiana 7-B.1 (2004): 61-77. <http://eudml.org/doc/196157>.

@article{CalderónMartín2004,
abstract = {Jordan $H^\{*\}$-pairs appear, in a natural way, in the study of Lie $H^\{*\}$-triple systems ([3]). Indeed, it is shown in [4, Th. 3.1] that the problem of the classification of Lie $H^\{*\}$-triple systems is reduced to prove the existence of certain $L^\{*\}$-algebra envelopes, and it is also shown in [3] that we can associate topologically simple nonquadratic Jordan $H^\{*\}$-pairs to a wide class of Lie $H^\{*\}$-triple systems and then the above envelopes can be obtained from a suitable classification, in terms of associative $H^\{*\}$-pairs, of these pairs. In this paper we give a classification theorem for topologically simple non-quadratic Jordan $H^\{*\}$-pairs in terms of associative $H^\{*\}$-pairs and certain of their anti-isomorphisms. Some consequences of this classification are also stated.},
author = {Calderón Martín, A. J., González, C. Martín},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {61-77},
publisher = {Unione Matematica Italiana},
title = {A structure theory for Jordan $H^*$-pairs},
url = {http://eudml.org/doc/196157},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Calderón Martín, A. J.
AU - González, C. Martín
TI - A structure theory for Jordan $H^*$-pairs
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/2//
PB - Unione Matematica Italiana
VL - 7-B
IS - 1
SP - 61
EP - 77
AB - Jordan $H^{*}$-pairs appear, in a natural way, in the study of Lie $H^{*}$-triple systems ([3]). Indeed, it is shown in [4, Th. 3.1] that the problem of the classification of Lie $H^{*}$-triple systems is reduced to prove the existence of certain $L^{*}$-algebra envelopes, and it is also shown in [3] that we can associate topologically simple nonquadratic Jordan $H^{*}$-pairs to a wide class of Lie $H^{*}$-triple systems and then the above envelopes can be obtained from a suitable classification, in terms of associative $H^{*}$-pairs, of these pairs. In this paper we give a classification theorem for topologically simple non-quadratic Jordan $H^{*}$-pairs in terms of associative $H^{*}$-pairs and certain of their anti-isomorphisms. Some consequences of this classification are also stated.
LA - eng
UR - http://eudml.org/doc/196157
ER -

References

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  1. ANQUELA, J. A.- CORTÉS, T., Primitive Jordan Pairs and Triple Systems, J. Algebra, 184, no. 2 (1996), 632-678. Zbl0857.17032MR1409234
  2. CALDERÓN, A. J.- MARTÍN, C., Dual pairs techniques in H * -theories, J. Pure Appl. Algebra, 133 (1998), 59-63. Zbl0962.17021MR1653695
  3. CALDERÓN, A. J.- MARTÍN, C., On L * -triples and Jordan H * -pairs, Ring theory and Algebraic Geometry, (Granja, Hermida, Verschoren eds.) Marcel Dekker, Inc. Chapter 4 (2001), 87-94. MR1844085
  4. CALDERÓN, A. J.- MARTÍN, C., Hilbert space methods in the theory of Lie triple systems, Recent Progress in Functional Analysis, K. D. Bierstedt, J. Bonet, M. Maestre, J. Schmets (ed.) in the series North-Holland Math. Studies (2001), 309-319. MR1861767
  5. CALDERÓN, A. J.- MARTÍN, C., On Associative and Jordan H * -pairs, Int. J. Math. Game Theory Algebra, 11, no. 4 (2001), 1-12. Zbl1034.46048MR1859391
  6. CASTELLÓN, A.- CUENCA, J. A., Isomorphisms of H * -triple systems, Ann. della Scuola Norm. Sup. Pisa Cl. Sci. 4, no. 4 (1992), 507-514. Zbl0805.46055MR1205882
  7. CASTELLÓN, A.- CUENCA, J. A., Associative H * -triple systems, In Nonassociative Algebraic Models. Nova Science Publishers, S. González and H.C. Myung Eds. (1992), 45-67. Zbl0794.46044MR1189612
  8. CASTELLÓN, A.- CUENCA, J. A., The Centroid and Metacentroid of an H * -triple system., Bull. Soc. Math. Belg, 45, Fac. 1 et 2 (1993), 85-93. MR1316233
  9. CASTELLÓN, A.- CUENCA, J. A., Jordan H * -triple systems, in Nonassociative Algebras and its Applications, S. González editor, Kluwer Academic Publishers (1994), 66-72. MR1338159
  10. CASTELLÓN, A.- CUENCA, J. A.- MARTÍN, C., Ternary H * -algebras, Boll. Un. Mat. Ital. B (7), 6, no. 1 (1992), 217-228. MR1164947
  11. CASTELLÓN, A.- CUENCA, J. A.- MARTÍN, C., Special Jordan H * -triple systems, Comm. Alg, 28, no. 10 (2000), 4699-4706. MR1779866
  12. CUENCA, J. A.- GARCÍA, A.- MARTÍN, C.. Jacobson density for associative pairs and its applications, Comm. Alg., 17, no. 10 (1989), 2595-2610. Zbl0694.17001MR1019184
  13. D'AMOUR, A., Jordan triple homomorphisms of associative structures, Comm. Algebra, 19, no. 4 (1991), 1229-1247. Zbl0728.17018MR1102336
  14. D'AMOUR, A., Zel'manov polynomials in quadratic Jordan triple systems, J. Algebra, 140, no. 1 (1991), 160-183. Zbl0796.17031MR1114912
  15. FERNÁNDEZ, A.- GARCÍA, E.- SÁNCHEZ, E., Prime Nondegenerate Jordan Triple Systems with Minimal Inner Ideals, Nonassociative algebraic modelsNova Sci. Publ., Commack, NY. (1992), 143-166. Zbl0769.17022MR1189618
  16. JACOBSON, N., Structure of Rings, American Mathematical Society Colloquium Publications vol. 37, 2nd ed. Providence R.I. Zbl0073.02002MR81264
  17. KAUP, W., Über die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension, I, Math. Ann., 257 (1981), 363-486. Zbl0482.32010MR639580
  18. KAUP, W., Über die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension, II, Math. Ann., 262 (1983), 57-75. Zbl0482.32011MR690007
  19. LOOS, O., On the socle of a Jordan pair, Collect. Math, 40, no. 2 (1989), 109-125. Zbl0729.17022MR1094683
  20. LOOS, O., Jordan pairs, Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, vol. 460, 1975. Zbl0301.17003MR444721
  21. MCCRIMMON, K.- ZEL'MANOV, E., The Stucture of Strongly Prime Quadratic Jordan Algebras, Adv. in Math, 69, no. 2 (1988), 133-222. Zbl0656.17015
  22. NEHER, E., Cartan-Involutionen von halbeinfachen rellen Jordan Triplesystemen, Math. Z, 169, no. 2 (1979), 271-292. Zbl0403.17014MR554530
  23. NEHER, E., On the classification of Lie and Jordan triple systems, Comm. Algebra, 13, no. 12 (1985), 2615-2667. Zbl0583.17001MR811526
  24. RODRIGUEZ, A., Jordan axioms for C * -algebras, Manuscripta Math., 61 (1988), 297-314. Zbl0665.46056MR949820

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