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APPROXIMATELY BIPROJECTIVE BANACH ALGEBRAS AND NILPOTENT IDEALS

Part of: Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  22 May 2012

HASAN POURMAHMOOD-AGHABABA
HASAN POURMAHMOOD-AGHABABA*
Affiliation:
Department of Mathematics, University of Tabriz, Tabriz, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box 19395-5746, Tehran, Iran (email: h_p_aghababa@tabrizu.ac.ir, h_pourmahmood@yahoo.com)
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Abstract

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By introducing a new notion of approximate biprojectivity we show that nilpotent ideals in approximately amenable or pseudo-amenable Banach algebras, and nilpotent ideals with the nilpotency degree larger than two in biflat Banach algebras cannot have the special property which we call ‘property (𝔹)’ (Definition 5.2 below) and hence, as a consequence, they cannot be boundedly approximately complemented in those Banach algebras.

Keywords

Banach algebrasbounded approximately complemented subspacesnilpotent idealsapproximate amenabilitypseudo-amenabilityapproximate biprojectivity

MSC classification

Secondary: 46H20: Structure, classification of topological algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 1 , February 2013 , pp. 158 - 173
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Aristov, O. Yu., ‘On approximation of flat Banach modules by free modules’, Sbornik. Math. 196(11) (2005), 1553–1583.CrossRefGoogle Scholar
[2]Curtis, P. C. Jr and Loy, R. J., ‘The structure of amenable Banach algebras’, J. Lond. Math. Soc. 40(2) (1989), 89–104.CrossRefGoogle Scholar
[3]Dales, H. G., Ghahramani, F. and Helemskii, A. Yu., ‘The amenability of measure algebras’, J. Lond. Math. Soc. 66(1) (2002), 213–226.CrossRefGoogle Scholar
[4]Dales, H. G., Lau, A. T. and Strauss, D., ‘Banach algebras on semigroups and on their compactifications’, Mem. Amer. Math. Soc. 205 (2010), 1–165.Google Scholar
[5]Dales, H. G., Loy, R. J. and Zhang, Y., ‘Approximate amenability for Banach sequence algebras’, Studia Math. 177 (2006), 81–96.CrossRefGoogle Scholar
[6]Forrest, B. E. and Runde, V., ‘Amenability and weak amenability of the Fourier algebra’, Math. Z. 250(4) (2005), 731–744.CrossRefGoogle Scholar
[7]Ghahramani, F. and Loy, R. J., ‘Generalized notions of amenability’, J. Funct. Anal. 208 (2004), 229–260.CrossRefGoogle Scholar
[8]Ghahramani, F., Loy, R. J. and Zhang, Y., ‘Generalized notions of amenability, II’, J. Funct. Anal. 254 (2008), 1776–1810.CrossRefGoogle Scholar
[9]Ghahramani, F. and Stokke, R., ‘Approximate and pseudo-amenability of the Fourier algebra’, Indiana Univ. Math. J. 56 (2007), 909–930.CrossRefGoogle Scholar
[10]Ghahramani, F. and Zhang, Y., ‘Pseudo-amenable and pseudo-contractible Banach algebras’, Math. Proc. Cambridge Philos. Soc. 142 (2007), 111–123.CrossRefGoogle Scholar
[11]Haagerup, U., ‘An example of a nonnuclear C*-algebra, which has the metric approximation property’, Invent. Math. 50 (1978/79), 279–293.CrossRefGoogle Scholar
[12]Helemskii, A. Ya., The Homology of Banach and Topological Algebras (Kluwer, Dordrecht, 1989).CrossRefGoogle Scholar
[13]Loy, R. J. and Willis, G. A., ‘The approximation property and nilpotent ideals in amenable Banach algebras’, Bull. Aust. Math. Soc. 49 (1994), 341–346.CrossRefGoogle Scholar
[14]Ramsden, P., ‘Biflatness of semigroup algebras’, Semigroup Forum 79 (2009), 515–530.CrossRefGoogle Scholar
[15]Runde, V., Lectures on Amenability, Lecture Notes in Mathematics, 1774 (Springer, Berlin, 2002).CrossRefGoogle Scholar
[16]Semei, E., Spronk, N. and Stokke, R., ‘Biflatness and Pseudo-Amenability of Segal algebras’, Canad. J. Math. 62(4) (2010), 845–869.CrossRefGoogle Scholar
[17]Zhang, Y., ‘Nilpotent ideals in a class of Banach algebras’, Proc. Amer. Math. Soc. 127(11) (1999), 3237–3242.CrossRefGoogle Scholar
[18]Zhang, Y., ‘Approximate complementation and its application in studying ideals of Banach algebras’, Math. Scand. 92 (2003), 301–308.CrossRefGoogle Scholar