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The approximation property and nilpotent ideals in amenable Banach algebras

Published online by Cambridge University Press:  17 April 2009

R.J. Loy  and
G.A. Willis
R.J. Loy
Affiliation:
Department of MathematicsSchool of Mathematical Sciences Australian National UniversityAustralian Capital Territory 0200
G.A. Willis
Affiliation:
Department of MathematicsThe University of NewcastleNew South Wales 2308, Australia
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It is well known that non-zero nilpotent ideals in amenable Banach algebras must be infinite-dimensional. We show that under certain additional hypotheses such ideals cannot even have the approximation property.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 49 , Issue 2 , April 1994 , pp. 341 - 346
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Bade, W.G. and Dales, H.G., ‘The Wedderburn decomposability of some commutative Banach algeras’, J. Funct. Anal. 107 (1992), 105–121.CrossRefGoogle Scholar
[2]Bonsall, F.F. and Duncan, J., Complete normed algebras (Springer-Verlag, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
[3]Curtis, P.C. Jr., ‘Complementation problems concerning the radical of a commutative amenable Banach algebra’, Proc. Centre Math. Anal. Austral. Nat. Univ. 21 (1989), 56–60.Google Scholar
[4]Curtis, P.C. Jr., and Loy, R.J., ‘The structure of amenable Banach algebras’, J. London Math. Soc. (2) 40 (1989), 89–104.CrossRefGoogle Scholar
[5]Grønbæk, N., Johnson, B.E. and Willis, G.A., ‘Amenability of Banach algebras of compact operators’, preprint (1992).Google Scholar
[6]Grønbæk, N. and Willis, G.A., ‘Approximate identities in Banach algebras of compact operators’, Canad. Math. Bull. 36 (1993), 45–53.CrossRefGoogle Scholar
[7]Helemskii, A.Ya., The homology of Banach and topological algebras (Kluwer, Dordrecht, 1986).Google Scholar
[8]Helson, H., ‘On the ideal structure of group algebras’, Ark. Mat. 2 (1952), 83–86.CrossRefGoogle Scholar
[9]Johnson, B.E., ‘Cohomology in Banach algebras’, Mem. Amer. Math. Soc. No. 127 (1972), 1–96.Google Scholar
[10]Selivanov, Y.V., ‘Projectivity of certain Banach modules and the structure of Banach algebras’, Soviet Math. (Iz. VUZ) 22 (1978), 88–93.Google Scholar
[11]Selivanov, Y.V., ‘Biprojective Banach algebras’, Izv. Akad. Nauk SSSR Ser. Matem. 43 (1979), 1159–1174.Google Scholar
[12]Selivanov, Y.V., ‘Homological characterizations of the approximation property for Banach spaces’, Glasgow Math. J. 34 (1992), 229–239.CrossRefGoogle Scholar
[13]Taylor, J.L., ‘Homology and cohomology for topological algebras’, Adv. in Math. 9 (1972), 137–182.CrossRefGoogle Scholar
[14]Varopoulos, N.Th., ‘Spectral synthesis on spheres’, Proc. Camb. Philos. Soc. 62 (1966), 379–387.CrossRefGoogle Scholar