An approach to Jordan-Banach algebras from the theory of nonassociative complete normed algebras

Angel Rodriguez Palacios

Annales scientifiques de l'Université de Clermont. Mathématiques (1991)

  • Volume: 97, Issue: 27, page 1-57
  • ISSN: 0249-7042

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Rodriguez Palacios, Angel. "An approach to Jordan-Banach algebras from the theory of nonassociative complete normed algebras." Annales scientifiques de l'Université de Clermont. Mathématiques 97.27 (1991): 1-57. <http://eudml.org/doc/80586>.

@article{RodriguezPalacios1991,
author = {Rodriguez Palacios, Angel},
journal = {Annales scientifiques de l'Université de Clermont. Mathématiques},
keywords = {survey; Jordan-Banach algebras; complete normed algebras; -algebras; nonassociative -algebras},
language = {eng},
number = {27},
pages = {1-57},
publisher = {UER de Sciences exactes et naturelles de l'Université de Clermont},
title = {An approach to Jordan-Banach algebras from the theory of nonassociative complete normed algebras},
url = {http://eudml.org/doc/80586},
volume = {97},
year = {1991},
}

TY - JOUR
AU - Rodriguez Palacios, Angel
TI - An approach to Jordan-Banach algebras from the theory of nonassociative complete normed algebras
JO - Annales scientifiques de l'Université de Clermont. Mathématiques
PY - 1991
PB - UER de Sciences exactes et naturelles de l'Université de Clermont
VL - 97
IS - 27
SP - 1
EP - 57
LA - eng
KW - survey; Jordan-Banach algebras; complete normed algebras; -algebras; nonassociative -algebras
UR - http://eudml.org/doc/80586
ER -

References

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  1. 1 A.A. Albert, The radical of a non-associative algebra, Bull. Amer. Math. Soc.48 (1942), 891-897. Zbl0061.05001MR7396
  2. 2 A.A. Albert, Absolute valued real algebras, Ann. of Math.48 (1947), 495-501. Zbl0029.01001MR20550
  3. 3 A.A. Albert, Absolute valued algebraic algebras, Bull. Amer. Math. Soc.55 (1949), 763-768. Zbl0033.34901MR30941
  4. 4 E.M. Alfsen, Compact convex sets and boundary Integrals, Erg. der Math.57, Springer-Verlag, Berlin1971. Zbl0209.42601MR445271
  5. 5 E.M. Alfsen, H. Hanche-Olsen and F.W. Shultz, State spaces of C*-algebras, Acta Math.144 (1980), 267-305. Zbl0458.46047MR573454
  6. 6 E.M. Alfsen, F.W. Shultz and E. Stormer, A Gelfand Neumark theorem for Jordan algebras, Adv. Math.28 (1978), 11-56. Zbl0397.46065MR482210
  7. 7 K. Alvermann and G. Janssen, Real and complex noncommutative Jordan Banach algebras, Math. Z.185 (1984), 105-113. Zbl0513.46044MR724047
  8. 8 C. Aparicio, F. Ocana, R. Paya and A. Rodriguez, A non-smooth extension of Frechet differentiability of the norm with applications to numerical ranges, Glasgow Math. J.28 (1986), 121-137. Zbl0604.46021MR848419
  9. 9 C. Aparicio and A. Rodriguez, Sobre el espectro de derivationes y automorfismos de las algebras de Banach, Rev. Real Acad. Cienc. Exactas, Fisicas y Naturales de Madrid79 (1985), 113-118. MR835174
  10. 10 B. Aupetit, The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras, J. Funct. Anal.47 (1982), 7-25. Zbl0488.46043MR663830
  11. 11 J.M. BACHAR, W.G. BADE, P.C. CURTUS, H.G. DALES and M.P. THOMAS (Editors), Radical Banach Algebras and Automatic Continuity, Lect. Notes in Math.975, Springer-Verlag, Berlin1983. Zbl0492.00008MR697577
  12. 12 G.F. Bachelis, Homomorphisms of annihilator Banach algebras, Pacific J. Math.25 (1968), 229-247. Zbl0164.15701MR244762
  13. 13 W.G. Bade and P.C. Curtis, The continuity of derivations of Banach algebras, J. Funct. Anal.16 (1974), 372-387. Zbl0296.46049MR358354
  14. 14 V.K. Balachandran and P.S. Rema, Uniqueness of the norm topology in certain Banach Jordan algebras, Publ. Ramanujan Inst.1 (1968/69), 283-289. Zbl0205.42204MR270153
  15. 15 T. Barton and R.M. Timoney, On biduals, preduals, and ideals of JB*-triples, University of California Irvine. Math. Scand.59 (1986), 177-191. Zbl0621.46044MR884654
  16. 16 F.F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras, Lecture Notes Series 2, London Math. Soc., Cambridge1971. Zbl0207.44802MR288583
  17. 17 F.F. Bonsall and J. Duncan, Numerical ranges II, Lecture Notes Series 10, London Math. Soc. Cambridge1973. Zbl0262.47001MR442682
  18. 18 F.F. Bonsall and J. Duncan, Complete Normed Algebras, Erg. der Math.80, Springer-Verlag, Berlin1973. Zbl0271.46039MR423029
  19. 19 R.B. Braun, Structure and representations of noncommutative C*-Jordan algebras, Manuscripta Math.41 (1983), 139-171. Zbl0512.46055MR689135
  20. 20 R.B. Braun, A Gelfand-Neumark theorem for C*-alternative algebras, Math. Z.185 (1984), 225-242. Zbl0514.46047MR731343
  21. 21 L.J. Bunce, The theory and structure of dual JB-algebras, Math. Z.180 (1982), 525-534. Zbl0518.46053MR667006
  22. 22 P. Civin and B. Yood, Lie and Jordan structures in Banach algebras, Pacific J. Math.15 (1965), 775-797. Zbl0135.35701MR188813
  23. 23 J.A. Cuenca, Sur la théorie des structures des H*-algebres de Jordan non commutatives. Colloque sur les algèbres de Jordan. Montpellier1985. 
  24. 24 J.A. Cuenca and A. Rodriguez, Isomorphism of H*-algebras, Math. Proc. Camb. Phil. Soc.97 (1985), 93-99. Zbl0571.46036MR764497
  25. 25 J.A. Cuenca and A. Rodriguez, Structure theory for noncommutative Jordan H*-algebras, J. Algebra, 106 (1987), 1-14. Zbl0616.46047MR878465
  26. 26 S. Dineen, The second dual of a JB*-triple system. Complex analysis, functional analysis and approximation theory (Campinas1984), 67-69, North-Holland Math. Stud., 125, North-Holland, Amsterdam-New York, 1986. Zbl0653.46053MR893410
  27. 27 J. Dixmier, Les algèbres d'opérateurs dans l'espace Hilbertien, Deuxième édition. Gauthier-Villars, Paris1969. Zbl0088.32304
  28. 28 A. Fernandez, Noncommutative Jordan algebras with minimal inner ideals. Colloque sur les algèbres de Jordan, Montpellier1985. 
  29. 29 A. Fernandez and A. Rodriguez, Primitive noncommutative Jordan algebras with nonzero socle, Proc. Amer. Math. Soc.96 (1986), 199-206. Zbl0585.17001MR818443
  30. 30 A. Fernandez and A. Rodriguez, A Wedderburn theorem for nonassociative complete normed algebras, J. London Math. Soc.33 (1986), 328-338. Zbl0603.46056MR838645
  31. 31 A. Fernandez and A. Rodriguez, Structure theory in alternative Banach algebras, Universidad de Granada. Preprint 1984. 
  32. 32 J.R. Giles, D.A. Gregory and B. Sims, Geometrical implications of upper semicontinuity of the duality mapping on a Banach space, Pacific J. Math.70 (1978), 99-108. Zbl0399.46012MR526669
  33. 33 H. Hanche-Olsen and E. Stormer, Jordan Operators Algebras, Monographs and Studies in Mathematics21, Pitman, 1984. Zbl0561.46031MR755003
  34. 34 P. de la HARPE, Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert space, Lect. Notes in Math.285, Springer-Verlag, Berlin1972. Zbl0256.22015MR476820
  35. 35 I.N. Hersttin, Topics in ring theory, Chicago, Univ. of Chicago Press1965. Zbl0232.16001
  36. 36 B. Iochum, Cônes autopolaires et algèbres de Jordan, Lect. Notes in Math.1049, Springer-Verlag, Berlin1984. Zbl0556.46040MR764767
  37. 37 B. Iochum and G. Loupias, Banach-power-associative algebras and Jordan algebras, Ann. Inst. H. Poincaré43 (1985), 211-225. Zbl0587.17012MR817536
  38. 38 N. Jacobson, Structure of rings, Amer. Math. Soc. Coll. Publ. 37, Providence, R.I., Amer. Math. Soc.1968. Zbl0073.02002MR222106
  39. 39 B.E. Johnson, A commutative semisimple annihilator Banach algebra which is not dual, Bull. Amer. Math. Soc.73 (1967), 407-409. Zbl0154.38703MR206739
  40. 40 B.E. Johnson and A.M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math.90 (1968), 1067-1073. Zbl0179.18103MR239419
  41. 41 R.V. Kadison, A representation theory for commutative topological algebras, Mem. Amer. Math. Soc.7 (1951). Zbl0042.34801MR44040
  42. 42 A.M. Kaidi, J. Martinez and A. Rodriguez, On a nonassociative Vidav-Palmer theorem, Quart. J. Math. Oxford (2), 32 (1981), 435-442. Zbl0446.46043MR635592
  43. 43 W. Kaup, Algebraic characterization of symmetric complex Banach manifolds, Math. Ann.228 (1977), 39-64. Zbl0335.58005MR454091
  44. 44 E. Kleinfeld, Primitive alternative rings and semisimplicity, Ann. of Math.77 (1955), 725-730. Zbl0066.02302MR72115
  45. 45 E. Kleinfeld, M. Kleinfeld and F. Kosier, A generalization of commutative and alternative rings, Canad. J. Math.22 (1970), 348-362. Zbl0198.35503MR258896
  46. 46 D. Li, Espaces L-facteurs de leurs biduaux: bonne disposition, meilleure approximation et propriété de Radon-Nikodym. Quart.J. Math. Oxford Ser.38 (1987), 229-243. Zbl0631.46020MR891618
  47. 47 G. Lumer, Complex methods and the estimations of operator norms and spectra from real numerical ranges, J. Funct. Anal.10 (1972), 482-495. Zbl0252.47002MR448122
  48. 48 J. Martinez, Holomorphic functional calculus in Jordan-Banach algebras, Colloque sur les algèbres de Jordan. Montpellier1985. 
  49. 49 J. Martinez, JV-algebras, Math. Proc. Camb. Phil. Soc.87 (1980), 47-50. Zbl0425.46037MR549296
  50. 50 J. Martinez, J.F. Mena, R. Paya and A. Rodriguez, An approach to numerical ranges without Banach algebra theory, Illinois, J. Math.29 (1985), 609-635. Zbl0604.46052MR806469
  51. 51 K McCrimmon, A general theory of Jordan rings, Proc. Nat. Acad. Sci. USA56 (1966), 1072-1079. Zbl0139.25502MR202783
  52. 52 K McCrimmon, The radical of a Jordan algebra, Proc. Nat. Acad. Sci. USA62 (1969), 671-678. Zbl0175.31002MR268238
  53. 53 K McCrimmon, On Hertein's theorems relating Jordan and associative algebras, J. Algebra13 (1969), 382-392. Zbl0224.16027MR249476
  54. 54 K McCrimmon, Noncommutative Jordan rings, Trans. Amer. Math. Soc.158 (1971), 1-33. Zbl0229.17002MR310024
  55. 55 J.F. Mena, R. Paya and A. Rodriguez, Semisummands and semiideals in Banach spaces, Israel J. Math.51 (1985), 33-67. Zbl0615.46018MR804475
  56. 56 J.I. Nieto, Gateaux differentials in Banach algebras, Math. Z.139 (1974), 23-34. Zbl0275.46036MR358353
  57. 57 J.M. Osborn and M.L. Racine, Jordan rings with nonzero socle, Trans. Amer. Math. Soc.251 (1979), 375-387. Zbl0409.17014MR531985
  58. 58 R. Paya, J. Perez and A. Rodriguez, Noncommutative Jordan C*-algebras, Muniscripta Math.37 (1982), 87-120. Zbl0483.46049MR649567
  59. 59 R. Paya, J., Perez and A. Rodriguez, Type I factor representations of noncommutative JB*- algebras, Proc. London Math. Soc. (3) 48 (1984), 428-444. Zbl0509.46052MR735223
  60. 60 PEREZ de GUZMAN, Structure theorems for alternative H*-algebras, Math. Proc. Camb. Phil. Soc.94 (1983), 437-446. Zbl0551.46037MR720794
  61. 61 P.S. Putter and B. Yood, Banach-Jordan *-algebras, Proc. London. Math. Soc. (3) 41.(1980), 21-44. Zbl0387.46048MR579715
  62. 62 A. Rodriguez, A Vidav-Palmer theorem for Jordan C*-algebras and related topics, J. London Math. Soc. (2) 22 (1980), 318-332. Zbl0483.46050MR588278
  63. 63 A. Rodriguez, Non-associative normed algebras spanned by hermitian elements, Proc. London Math. Soc. (3) 47 (1983), 258-274. Zbl0521.47036MR703979
  64. 64 A. Rodriguez, The Uniqueness of the complete norm topology in complete normed nonassociative algebras, J. Funct. Anal.60 (1985), 1-15. Zbl0602.46055MR780101
  65. 65 S. Sakai, C*- algebras and W*-algebras, Erg. der Math.60, Springer-Verlag, Berlin1971. Zbl0219.46042MR442701
  66. 66 R.D. Schafer, Generalized standard algebras, J. Algebra12 (1969) 376-417. Zbl0196.06102MR283035
  67. 67 J.T. Schue, Hilbert space methods in the theory of Lie algebras, Trans. Amer. Math. Soc.95 (1960), 69-80. Zbl0093.30601MR117575
  68. 68 J.R. Schue, Cartan deocmposition for L*-algebras, Trans. Amer; Math. Soc.98 (1961), 334-349. Zbl0099.10205MR133408
  69. 69 A.M. Sinclair, Jordan homomorphism and derivations on a semisimple Banach algebra, Proc. Amer. Math. Soc.24 (1970), 209-214. Zbl0175.44001MR250069
  70. 70 A.M. Sinclair, Automatic continuity of linear operators, London Math. Soc. Letc. Notes Series 21, Cambridge U. Press1976. Zbl0313.47029MR487371
  71. 71 M.F. Smiley, The radical of an alternative ring, Ann. of Math.49 (1948), 702-709. Zbl0031.34501MR25449
  72. 72 R.R. Smith, The numerical range in the second dual of a Banach algebra, Math. Proc. Camb. Phil. Soc.89 (1981), 301-307. Zbl0461.46035MR600245
  73. 73 E. Strzelecki, Power-associative regular real normed algebras, J. Austral. Math. Soc.6 (1966), 193-209. Zbl0145.16504MR198276
  74. 74 H. Upmeier, Symmetric Banach Manifolds and Jordan C*-algebras, North-Holland Mathematics Studies104, Elsevier Science Publishers B.V., Amsterdam1985. Zbl0561.46032MR776786
  75. 75 K Urbanik and F.B. Wright, Absolute-valued algebras, Proc. Amer. Math. Soc.11 (1960), 861-866. Zbl0156.03801MR120264
  76. 76 E. Vesentini, On the subharmonicity of the spectral radius, Boll. Un. Mat. Ital.4 (1968), 427-429. MR244766
  77. 77 C. Viola, Hilbert space methods in the theory of Jordan algebras, Math. Proc. Camb. Phil. Soc.78 (1974), 293-300. Zbl0357.17015
  78. 78 C. Viola and P.S. Rema, Hilbert space methods in the theory of Jordan algebras II, Math. Proc. Camb. Phil. Soc.79 (1976), 307-319. Zbl0357.17016MR404360
  79. 79 F.B. Wright, Absolute valued algebras, Proc. Nat. Acad. Sci. USA39 (1953), 330-332. Zbl0050.03103MR54583
  80. 80 J.D.M. Wright, Jordan C*-algebras, Michigan Math. J.24 (1977), 291-302. Zbl0384.46040MR487478
  81. 81 J.D.M. Wright and M.A. Yougson, On isometries of Jordan algebras, J. London Math. Soc. (2) 17 (1978), 339-344. Zbl0384.46041MR482212
  82. 82 B. Yood, Closed prime ideals in topological rings, Proc. London Math. Soc.24 (1972), 307-323. Zbl0232.46041MR298423
  83. 83 B. Yood, Homomorphisms on normed algebras, Pacific J. Math.8 (1958), 373-381. Zbl0084.33601MR104164
  84. 84 M.A. Yougson, A Vidav theorem for Banach Jordan algebras, Math. Proc. Camb. Phil. Soc.84 (1978), 263-272. Zbl0392.46038MR493372
  85. 85 M.A. Yougson, Hermitian operators on Banach Jordan algebras, Proc. Edinburgh Math. Soc. (2) 22 (1979), 93-104. Zbl0414.46034MR549462
  86. 86 M.A. Yougson, Non-unital Banach Jordan algebras and C*-triple systems, Proc. Edinburgh Math. Soc. (2) 24 (1981), 19-30. Zbl0441.46042
  87. 87 K.A. Zhevlakov, Coincidence of Smiley and Kleinfeld radicals in alternative rings, Algebra i Logika8 (1969), 309-319. Zbl0214.05201MR266972
  88. 88 D.P. Blecher, Z. Ruan and A.M. Sinclair, A characterization of operator algebras, J. Funct. Anal.89 (1990), 188-201. Zbl0714.46043MR1040962
  89. 89 M. Bresar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc.104 (1988), 1003-1006. Zbl0691.16039MR929422
  90. 90 M. Cabrera, J. Martinez and A. Rodriguez, Structurable H*-algebras, J. Algebra (to appear). Zbl0749.17001
  91. 91 M. Cabrera and A. Rodriguez, Extended centroid and central closure of semiprime normed algebras: a first approach, Commun. Algebra18 (1990), 2293-2326. Zbl0713.46033MR1063136
  92. 92 M. Cabrera and A. Rodriguez, Nonassociative ultraprime normed algebras, Quart. J. Math. Oxford (to appear). Zbl0758.46037
  93. 93 J.M. Cusack, Jordan derivations on rings, Proc. Amer; Math. Soc.53 (1975), 321-324. Zbl0327.16020MR399182
  94. 94 T.S. Erickson, W.S. Martindale 3rd and J.M. Osborn, Prime nonassociative algebras, Pacific J. Math.60 (1975), 49-63. Zbl0355.17005MR382379
  95. 95 A. Fernandez, E. Garcia and A. Rodrigurez, A Zelmanov prime theorem for JB*-algebras, J. London Math. Soc. (to appear). 
  96. 96 L. Hogben and K. McCrimmon, Maximal modula inner ideals and the Jacobson radical of a Jordan algebra, J. Algebra68 (1981), 156-169. Zbl0449.17011MR604300
  97. 97 M. Mathieu, Rings of quotients of ultraprime Banach algebras, with applications to elementary operators, Proc. Centre Math. Anal. Austral. Nat. Univ.21 (1989), 297-317. Zbl0701.46027MR1022011
  98. 98 A. Rodriguez, Primitive nonassociative normed algebras and extended centroid. In "Workshop on nonassociative algebraic models", Nova Science Publishers, New York (to appear). Zbl0941.46500MR1189625
  99. 99 A.R. Villena, Continuity of derivations on H*-algebras. Preprint 1991. Zbl0822.46061MR1207543
  100. 100 K.A. Zhevlakov, A.M. Slin'ko, I.P. Shestakov and A.I. Shirshov, Rings that are nearly associative, Acadepic Press1982. Zbl0487.17001MR668355

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