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Topologically simple Banach algebras with derivation

Published online by Cambridge University Press:  17 April 2009

El Hossein Illoussamen
Affiliation:
Ecole Normale Supérieure TakaddoumDépartement de MathématiquesB.P. 511810105 RabatMorocco
Volker Runde
Affiliation:
Fachbereich 9 MathematikUniversitát des SaarlandesPostfach 15115066041 SaarbrückenGermany e-mail: [email protected]
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Abstract

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It is not known if a commutative, topologically simple, radical Banach algebra exists. If, however, every derivation on such an algebra is continuous, this yields the automatic continuity of all derivations on commutative, semiprime Banach algebras. Utilising techniques used by Thomas in his proof of the Singer-Wermer conjecture, we show that, if A is a commutative, topologically simple Banach algebra with a non-zero derivation on it, then a quotient of a certain localisation of A has a power series structure. A pivotal role is played by what we call ample sets of denominators.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Allan, G. R., ‘Embedding the algebra of formal power series in a Banach algebra’, Proc. London Math. Soc. (3) 25 (1972), 329340.CrossRefGoogle Scholar
[2]Allan, G.R. Elements of finite closed descent in a Banach algebra, J. London Math. Soc. (2) 7 (1973), 462466.Google Scholar
[3]Atiyah, M.F. and MacDonald, I. G., Introduction to commutative algebra (Addison-Wesley, Massachusetts, London, Ontario, 1969).Google Scholar
[4]Curtis, P.C. Jr, ‘Derivations in commutative Banach algebras’, in Radical Banach algebras and automatic continuity, (Bachar, J.M. et al. , Editors), Springer Lecture Notes in Mathematics 975 (Springer Verlag, Berlin, Heidelberg, New York, 1983), pp. 328333.CrossRefGoogle Scholar
[5]Cusack, J., ‘Automatic continuity and topologically simple radical Banach algebras’, J. London Math. Soc. (2) 16 (1977), 493500.CrossRefGoogle Scholar
[6]Dixon, P.G., ‘Semiprime Banach algebras’, J. London Math. Soc. (2) 6 (1973), 676678.CrossRefGoogle Scholar
[7]Esterle, J., ‘Mittag-Leffler methods in the theory of Banach algebras and a new approach to Michael's problem’, in Proceedings of the Conference on Banach Algebras and Several Complex Variables, (Greenleaf, F. and Gulick, D., Editors), Contemporary Mathematics 32 (Amer. Math. Soc, Providence R.I., 1984), pp. 107129.CrossRefGoogle Scholar
[8]Garimella, R.V., ‘Continuity of derivations on some semiprime Banach algebras’, Proc. Amer. Math. Soc. 99 (1987), 289292.CrossRefGoogle Scholar
[9]Illoussamen, E., ‘Continuité des dérivations et des épimorphismes dans certaines algèbres de Banach’, Rend. Circ. Mat. Paĺermo (2) 44 (1995), 173186.CrossRefGoogle Scholar
[10]Johnson, B.E., ‘Continuity of derivations on commutative algebras’, Amer. J. Math. 91 (1969), 110.CrossRefGoogle Scholar
[11]Runde, V., ‘Automatic continuity of derivations and epimorphisms’, Pacific J. Math. 147 (1991), 365374.CrossRefGoogle Scholar
[12]Singer, I.M. and Wermer, J., ‘Derivations on commutative normed algebras’, Math. Ann. 129 (1955), 260264.CrossRefGoogle Scholar
[13]Suciu, I., ‘Eine natürliche Erweiterung der kommutativen Banachalgebren’, Rev. Roum. Math. Pures Appl. 7 (1962), 483491.Google Scholar
[14]Suciu, I., ‘Bruchalgebren der Banachalgebren’, Rev. Roum. Math. Pures Appl. 8 (1963), 313316.Google Scholar
[15]Thomas, M.P., ‘The image of a derivation is contained in the radical’, Ann. of Math. 128 (1988), 435460.CrossRefGoogle Scholar