An approach to Jordan-Banach algebras from the theory of nonassociative complete normed algebras
Annales scientifiques de l'Université de Clermont. Mathématiques, Volume 97 (1991) no. 27, pp. 1-57.
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     title = {An approach to {Jordan-Banach} algebras from the theory of nonassociative complete normed algebras},
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     volume = {97},
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     year = {1991},
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     zbl = {0768.17014},
     language = {en},
     url = {http://archive.numdam.org/item/ASCFM_1991__97_27_1_0/}
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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, Volume 97 (1991) no. 27, pp. 1-57. http://archive.numdam.org/item/ASCFM_1991__97_27_1_0/

1 A.A. Albert, The radical of a non-associative algebra, Bull. Amer. Math. Soc. 48 (1942), 891-897. | MR | Zbl

2 A.A. Albert, Absolute valued real algebras, Ann. of Math. 48 (1947), 495-501. | MR | Zbl

3 A.A. Albert, Absolute valued algebraic algebras, Bull. Amer. Math. Soc. 55 (1949), 763-768. | MR | Zbl

4 E.M. Alfsen, Compact convex sets and boundary Integrals, Erg. der Math. 57, Springer-Verlag, Berlin 1971. | MR | Zbl

5 E.M. Alfsen, H. Hanche-Olsen and F.W. Shultz, State spaces of C*-algebras, Acta Math. 144 (1980), 267-305. | MR | Zbl

6 E.M. Alfsen, F.W. Shultz and E. Stormer, A Gelfand Neumark theorem for Jordan algebras, Adv. Math. 28 (1978), 11-56. | MR | Zbl

7 K. Alvermann and G. Janssen, Real and complex noncommutative Jordan Banach algebras, Math. Z. 185 (1984), 105-113. | MR | Zbl

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. | MR | Zbl

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 Madrid 79 (1985), 113-118. | MR

10B. Aupetit, The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras, J. Funct. Anal. 47 (1982), 7-25. | MR | Zbl

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, Berlin 1983. | MR | Zbl

12G.F. Bachelis, Homomorphisms of annihilator Banach algebras, Pacific J. Math. 25 (1968), 229-247. | MR | Zbl

13W.G. Bade and P.C. Curtis, The continuity of derivations of Banach algebras, J. Funct. Anal. 16 (1974), 372-387. | MR | Zbl

14V.K. Balachandran and P.S. Rema, Uniqueness of the norm topology in certain Banach Jordan algebras, Publ. Ramanujan Inst. 1 (1968/69), 283-289. | MR | Zbl

15T. Barton and R.M. Timoney, On biduals, preduals, and ideals of JB*-triples, University of California Irvine. Math. Scand. 59 (1986), 177-191. | MR | Zbl

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., Cambridge 1971. | MR | Zbl

17 F.F. Bonsall and J. Duncan, Numerical ranges II, Lecture Notes Series 10, London Math. Soc. Cambridge 1973. | MR | Zbl

18F.F. Bonsall and J. Duncan, Complete Normed Algebras, Erg. der Math. 80, Springer-Verlag, Berlin 1973. | MR | Zbl

19R.B. Braun, Structure and representations of noncommutative C*-Jordan algebras, Manuscripta Math. 41 (1983), 139-171. | MR | Zbl

20R.B. Braun, A Gelfand-Neumark theorem for C*-alternative algebras, Math. Z. 185 (1984), 225-242. | MR | Zbl

21L.J. Bunce, The theory and structure of dual JB-algebras, Math. Z. 180 (1982), 525-534. | MR | Zbl

22P. Civin and B. Yood, Lie and Jordan structures in Banach algebras, Pacific J. Math. 15 (1965), 775-797. | MR | Zbl

23 J.A. Cuenca, Sur la théorie des structures des H*-algebres de Jordan non commutatives. Colloque sur les algèbres de Jordan. Montpellier 1985.

24J.A. Cuenca and A. Rodriguez, Isomorphism of H*-algebras, Math. Proc. Camb. Phil. Soc. 97 (1985), 93-99. | MR | Zbl

25J.A. Cuenca and A. Rodriguez, Structure theory for noncommutative Jordan H*-algebras, J. Algebra, 106 (1987), 1-14. | MR | Zbl

26S. Dineen, The second dual of a JB*-triple system. Complex analysis, functional analysis and approximation theory (Campinas 1984), 67-69, North-Holland Math. Stud., 125, North-Holland, Amsterdam-New York, 1986. | MR | Zbl

27 J. Dixmier, Les algèbres d'opérateurs dans l'espace Hilbertien, Deuxième édition. Gauthier-Villars, Paris 1969. | Zbl

28 A. Fernandez, Noncommutative Jordan algebras with minimal inner ideals. Colloque sur les algèbres de Jordan, Montpellier 1985.

29A. Fernandez and A. Rodriguez, Primitive noncommutative Jordan algebras with nonzero socle, Proc. Amer. Math. Soc. 96 (1986), 199-206. | MR | Zbl

30A. Fernandez and A. Rodriguez, A Wedderburn theorem for nonassociative complete normed algebras, J. London Math. Soc. 33 (1986), 328-338. | MR | Zbl

31 A. Fernandez and A. Rodriguez, Structure theory in alternative Banach algebras, Universidad de Granada. Preprint 1984.

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. | MR | Zbl

33H. Hanche-Olsen and E. Stormer, Jordan Operators Algebras, Monographs and Studies in Mathematics 21, Pitman, 1984. | MR | Zbl

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, Berlin 1972. | MR | Zbl

35 I.N. Hersttin, Topics in ring theory, Chicago, Univ. of Chicago Press 1965. | Zbl

36 B. Iochum, Cônes autopolaires et algèbres de Jordan, Lect. Notes in Math. 1049, Springer-Verlag, Berlin 1984. | MR | Zbl

37B. Iochum and G. Loupias, Banach-power-associative algebras and Jordan algebras, Ann. Inst. H. Poincaré 43 (1985), 211-225. | Numdam | MR | Zbl

38 N. Jacobson, Structure of rings, Amer. Math. Soc. Coll. Publ. 37, Providence, R.I., Amer. Math. Soc. 1968. | MR | Zbl

39B.E. Johnson, A commutative semisimple annihilator Banach algebra which is not dual, Bull. Amer. Math. Soc. 73 (1967), 407-409. | MR | Zbl

40B.E. Johnson and A.M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968), 1067-1073. | MR | Zbl

41R.V. Kadison, A representation theory for commutative topological algebras, Mem. Amer. Math. Soc. 7 (1951). | MR | Zbl

42A.M. Kaidi, J. Martinez and A. Rodriguez, On a nonassociative Vidav-Palmer theorem, Quart. J. Math. Oxford (2), 32 (1981), 435-442. | MR | Zbl

43W. Kaup, Algebraic characterization of symmetric complex Banach manifolds, Math. Ann. 228 (1977), 39-64. | MR | Zbl

44 E. Kleinfeld, Primitive alternative rings and semisimplicity, Ann. of Math. 77 (1955), 725-730. | MR | Zbl

45E. Kleinfeld, M. Kleinfeld and F. Kosier, A generalization of commutative and alternative rings, Canad. J. Math. 22 (1970), 348-362. | MR | Zbl

46D. 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. | MR | Zbl

47G. Lumer, Complex methods and the estimations of operator norms and spectra from real numerical ranges, J. Funct. Anal. 10 (1972), 482-495. | MR | Zbl

48 J. Martinez, Holomorphic functional calculus in Jordan-Banach algebras, Colloque sur les algèbres de Jordan. Montpellier 1985.

49J. Martinez, JV-algebras, Math. Proc. Camb. Phil. Soc. 87 (1980), 47-50. | MR | Zbl

50J. 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. | MR | Zbl

51K Mccrimmon, A general theory of Jordan rings, Proc. Nat. Acad. Sci. USA 56 (1966), 1072-1079. | MR | Zbl

52K Mccrimmon, The radical of a Jordan algebra, Proc. Nat. Acad. Sci. USA 62 (1969), 671-678. | MR | Zbl

53K Mccrimmon, On Hertein's theorems relating Jordan and associative algebras, J. Algebra 13 (1969), 382-392. | MR | Zbl

54K Mccrimmon, Noncommutative Jordan rings, Trans. Amer. Math. Soc. 158 (1971), 1-33. | MR | Zbl

55J.F. Mena, R. Paya and A. Rodriguez, Semisummands and semiideals in Banach spaces, Israel J. Math. 51 (1985), 33-67. | MR | Zbl

56J.I. Nieto, Gateaux differentials in Banach algebras, Math. Z. 139 (1974), 23-34. | MR | Zbl

57J.M. Osborn andM.L. Racine, Jordan rings with nonzero socle, Trans. Amer. Math. Soc. 251 (1979), 375-387. | MR | Zbl

58R. Paya, J. Perez and A. Rodriguez, Noncommutative Jordan C*-algebras, Muniscripta Math. 37 (1982), 87-120. | MR | Zbl

59R. Paya, J.,Perez and A. Rodriguez, Type I factor representations of noncommutative JB*- algebras, Proc. London Math. Soc. (3) 48 (1984), 428-444. | MR | Zbl

60 Perez De Guzman, Structure theorems for alternative H*-algebras, Math. Proc. Camb. Phil. Soc. 94 (1983), 437-446. | MR | Zbl

61P.S. Putter and B. Yood, Banach-Jordan *-algebras, Proc. London. Math. Soc. (3) 41.(1980), 21-44. | MR | Zbl

62A. Rodriguez, A Vidav-Palmer theorem for Jordan C*-algebras and related topics, J. London Math. Soc. (2) 22 (1980), 318-332. | MR | Zbl

63A. Rodriguez, Non-associative normed algebras spanned by hermitian elements, Proc. London Math. Soc. (3) 47 (1983), 258-274. | MR | Zbl

64A. Rodriguez, The Uniqueness of the complete norm topology in complete normed nonassociative algebras, J. Funct. Anal. 60 (1985), 1-15. | MR | Zbl

65S. Sakai, C*- algebras and W*-algebras, Erg. der Math. 60, Springer-Verlag, Berlin 1971. | MR | Zbl

66R.D. Schafer, Generalized standard algebras, J. Algebra 12 (1969) 376-417. | MR | Zbl

67J.T. Schue, Hilbert space methods in the theory of Lie algebras, Trans. Amer. Math. Soc. 95 (1960), 69-80. | MR | Zbl

68 J.R. Schue, Cartan deocmposition for L*-algebras, Trans. Amer; Math. Soc. 98 (1961), 334-349. | MR | Zbl

69A.M. Sinclair, Jordan homomorphism and derivations on a semisimple Banach algebra, Proc. Amer. Math. Soc. 24 (1970), 209-214. | MR | Zbl

70 A.M. Sinclair, Automatic continuity of linear operators, London Math. Soc. Letc. Notes Series 21, Cambridge U. Press 1976. | MR | Zbl

71M.F. Smiley, The radical of an alternative ring, Ann. of Math. 49 (1948), 702-709. | MR | Zbl

72R.R. Smith, The numerical range in the second dual of a Banach algebra, Math. Proc. Camb. Phil. Soc. 89 (1981), 301-307. | MR | Zbl

73E. Strzelecki, Power-associative regular real normed algebras, J. Austral. Math. Soc. 6 (1966), 193-209. | MR | Zbl

74H. Upmeier, Symmetric Banach Manifolds and Jordan C*-algebras, North-Holland Mathematics Studies 104, Elsevier Science Publishers B.V., Amsterdam 1985. | MR | Zbl

75K Urbanik andF.B. Wright, Absolute-valued algebras, Proc. Amer. Math. Soc. 11 (1960), 861-866. | MR | Zbl

76E. Vesentini, On the subharmonicity of the spectral radius, Boll. Un. Mat. Ital. 4 (1968), 427-429. | MR

77C. Viola, Hilbert space methods in the theory of Jordan algebras, Math. Proc. Camb. Phil. Soc. 78 (1974), 293-300. | Zbl

78C. Viola and P.S. Rema, Hilbert space methods in the theory of Jordan algebras II, Math. Proc. Camb. Phil. Soc. 79 (1976), 307-319. | MR | Zbl

79F.B. Wright, Absolute valued algebras, Proc. Nat. Acad. Sci. USA 39 (1953), 330-332. | MR | Zbl

80 J.D.M. Wright, Jordan C*-algebras, Michigan Math. J. 24 (1977), 291-302. | MR | Zbl

81 J.D.M. Wright and M.A. Yougson, On isometries of Jordan algebras, J. London Math. Soc. (2) 17 (1978), 339-344. | MR | Zbl

82B. Yood, Closed prime ideals in topological rings, Proc. London Math. Soc. 24 (1972), 307-323. | MR | Zbl

83B. Yood, Homomorphisms on normed algebras, Pacific J. Math. 8 (1958), 373-381. | MR | Zbl

84M.A. Yougson, A Vidav theorem for Banach Jordan algebras, Math. Proc. Camb. Phil. Soc. 84 (1978), 263-272. | MR | Zbl

85M.A. Yougson, Hermitian operators on Banach Jordan algebras, Proc. Edinburgh Math. Soc. (2) 22 (1979), 93-104. | MR | Zbl

86M.A. Yougson, Non-unital Banach Jordan algebras and C*-triple systems, Proc. Edinburgh Math. Soc. (2) 24 (1981), 19-30. | Zbl

87K.A. Zhevlakov, Coincidence of Smiley and Kleinfeld radicals in alternative rings, Algebra i Logika 8 (1969), 309-319. | MR | Zbl

88D.P. Blecher, Z. Ruan and A.M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1990), 188-201. | MR | Zbl

89M. Bresar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), 1003-1006. | MR | Zbl

90M. Cabrera, J. Martinez and A. Rodriguez, Structurable H*-algebras, J. Algebra (to appear). | Zbl

91M. Cabrera and A. Rodriguez, Extended centroid and central closure of semiprime normed algebras: a first approach, Commun. Algebra 18 (1990), 2293-2326. | MR | Zbl

92M. Cabrera and A. Rodriguez, Nonassociative ultraprime normed algebras, Quart. J. Math. Oxford (to appear). | Zbl

93 J.M. Cusack, Jordan derivations on rings, Proc. Amer; Math. Soc. 53 (1975), 321-324. | MR | Zbl

94T.S. Erickson, W.S. Martindale 3rd and J.M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), 49-63. | MR | Zbl

95A. Fernandez, E. Garcia and A. Rodrigurez, A Zelmanov prime theorem for JB*-algebras, J. London Math. Soc. (to appear).

96L. Hogben and K. Mccrimmon, Maximal modula inner ideals and the Jacobson radical of a Jordan algebra, J. Algebra 68 (1981), 156-169. | MR | Zbl

97M. Mathieu, Rings of quotients of ultraprime Banach algebras, with applications to elementary operators, Proc. Centre Math. Anal. Austral. Nat. Univ. 21 (1989), 297-317. | MR | Zbl

98 A. Rodriguez, Primitive nonassociative normed algebras and extended centroid. In "Workshop on nonassociative algebraic models", Nova Science Publishers, New York (to appear). | MR | Zbl

99 A.R. Villena, Continuity of derivations on H*-algebras. Preprint 1991. | MR | Zbl

100 K.A. Zhevlakov, A.M. Slin'Ko, I.P. Shestakov and A.I. Shirshov, Rings that are nearly associative, Acadepic Press 1982. | MR | Zbl