We establish that all derivations on a semisimple Jordan-Banach algebra are automatically continuous. By showing that "almost all" primitive ideals in the algebra are invariant under a given derivation, the general case is reduced to that of primitive Jordan-Banach algebras.
@article{bwmeta1.element.bwnjournal-article-smv118i3p205bwm,
author = {A. Villena},
title = {Derivations on Jordan-Banach algebras},
journal = {Studia Mathematica},
volume = {119},
year = {1996},
pages = {205-229},
zbl = {0855.46030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv118i3p205bwm}
}
Villena, A. Derivations on Jordan-Banach algebras. Studia Mathematica, Tome 119 (1996) pp. 205-229. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv118i3p205bwm/
[00000] [1] B. Aupetit, The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras, J. Funct. Anal. 47 (1982), 1-6. | Zbl 0488.46043
[00001] [2] M. Cabrera, A. Moreno and A. Rodríguez, Zel'manov's theorem for primitive Jordan-Banach algebras, J. London Math. Soc., to appear. | Zbl 0922.17019
[00002] [3] J. M. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 321-324.
[00003] [4] A. Fernández, Modular annihilator Jordan algebras, Comm. Algebra 13 (1985), 2597-2613. | Zbl 0613.17015
[00004] [5] L. Hogben and K. McCrimmon, Maximal modular inner ideals and the Jacobson radical of a Jordan algebra, J. Algebra 68 (1981), 155-169. | Zbl 0449.17011
[00005] [6] X. Jiang, Remarks on automatic continuity of derivations and module derivations, Acta Math. Sinica (N.S.) 4 (3) (1988) 227-233. | Zbl 0673.46026
[00006] [7] B. E. Johnson, The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537-539. | Zbl 0172.41004
[00007] [8] B. E. Johnson, Continuity of derivations on commutative algebras, Amer. J. Math. 91 (1969), 1-10. | Zbl 0181.41103
[00008] [9] B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, ibid. 90 (1968), 1067-1073. | Zbl 0179.18103
[00009] [10] K. B. Laursen, Some remarks on automatic continuity, in: Lecture Notes in Math. 512, Springer, 1976, 96-108.
[00010] [11] K. McCrimmon, The radical of a Jordan algebra, Proc. Nat. Acad. Sci. U.S.A. 59 (1969), 671-678. | Zbl 0175.31002
[00011] [12] K. McCrimmon and E. Zel'manov, The structure of strongly quadratic Jordan algebras, Adv. in Math. 69 (1988), 133-222. | Zbl 0656.17015
[00012] [13] A. Rodríguez, The uniqueness of the complete norm topology in complete normed nonassociative algebras, J. Funct. Anal. 60 (1985), 1-15. | Zbl 0602.46055
[00013] [14] A. Rodríguez, An approach to Jordan-Banach algebras from the theory of nonassociative complete normed algebras, Ann. Sci. Univ. Clermont-Ferrand II Math. 27 (1991), 1-57. | Zbl 0768.17014
[00014] [15] A. Rodríguez, Jordan structures in analysis, in: Jordan Algebras, Proc. Conf. Oberwolfach, August 9-15, 1992, W. Kaup, K. McCrimmon and H. Petersson (eds.), de Gruyter, Berlin, 1994, 97-186. | Zbl 0818.17036
[00015] [16] A. M. Sinclair, Jordan homomorphisms and derivations on semisimple Banach algebras, Proc. Amer. Math. Soc. 24 (1970), 209-214. | Zbl 0175.44001
[00016] [17] A. M. Sinclair, Continuity of Linear Operators, London Math. Soc. Lecture Note Ser. 21, Cambridge University Press, 1976. | Zbl 0313.47029
[00017] [18] M. P. Thomas, Primitive ideals and derivations on non-commutative Banach algebras, Pacific J. Math. 159 (1993), 139-152. | Zbl 0739.47014
[00018] [19] A. R. Villena, Continuity of derivations on a complete normed alternative algebra, J. Inst. Math. Comput. Sci. 3 (1990), 99-106. | Zbl 0876.46036
[00019] [20] A. R. Villena, Continuity of derivations on H*-algebras, Proc. Amer. Math. Soc. 122 (1994), 821-826. | Zbl 0822.46061
[00020] [21] E. I. Zel'manov, On prime Jordan algebras, Algebra i Logika 18 (1979), 162-175 (in Russian).
[00021] [22] E. I. Zel'manov, On prime Jordan algebras II, Siberian Math. J. 24 (1983), 89-104.