Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-12-01T08:59:49.532Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERALIZED AMENABILITY PROPERTIES OF THE BEURLING ALGEBRAS

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

Published online by Cambridge University Press:  01 April 2011

F. GHAHRAMANI ,
E. SAMEI  and
YONG ZHANG
F. GHAHRAMANI*
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, Canada R3T 2N2 (email: [email protected])
E. SAMEI
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada S7N 5E6 (email: [email protected])
YONG ZHANG
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, Canada R3T 2N2 (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that every one-codimensional closed two-sided ideal in a boundedly approximately contractible Banach algebra has a bounded approximate identity. We use this to give a complete characterization of bounded approximate contractibility of Beurling algebras associated to symmetric weights. We give a slight modification of a criterion for bounded approximate contractibility. We use our criterion to show that, for the quasi-SIN groups, in the presence of a certain growth condition on a weight, the associated Beurling algebra is boundedly approximately amenable if and only if it is boundedly approximately contractible. We show that approximate amenability of a Beurling algebra on an IN group necessitates the amenability of the group. Finally, we show that, for every locally compact abelian group, in the presence of a growth condition on the weight, 2n-weak amenability of the associated Beurling algebra is equivalent to every point-derivation vanishing at the augmentation character.

Keywords

amenable groupBeurling algebrabounded approximate amenabilitybounded approximate contractibilityone-codimensional idealIN groupquasi-SIN groupn-weak amenability

MSC classification

Secondary: 46H20: Structure, classification of topological algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 3 , December 2010 , pp. 359 - 376
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

Ghahramani was supported by NSERC Grant 36640-07, Samei was supported by NSERC Grant 3666066-09, Zhang was supported by NSERC Grant 238949-05.

References

[1]Aristov, O. Y., Runde, V. and Spronk, N., ‘Operator biflatness of the Fourier algebra and approximate indicators for subgroups’, J. Funct. Anal. 209 (2004), 367–387.CrossRefGoogle Scholar
[2]Choi, Y., Ghahramani, F. and Zhang, Y., ‘Approximate and pseudo-amenability of various classes of Banach algebras’, J. Funct. Anal. 256 (2009), 3158–3191.CrossRefGoogle Scholar
[3]Dales, H. G., Ghahramani, F. and Grønbæk, N., ‘Derivations into iterated duals of Banach algebras’, Studia Math. 128 (1998), 19–54.Google Scholar
[4]Dales, H. G. and Lau, A. T. M., ‘The second duals of Beurling algebras’, Mem. Amer. Math. Soc. 177(836) (2005).Google Scholar
[5]Ghahramani, F. and Lau, A. T., ‘Isometric isomorphisms between the second conjugate algebras of group algebras’, Bull. Lond. Math. Soc. 20 (1988), 342–344.CrossRefGoogle Scholar
[6]Ghahramani, F. and Loy, R. J., ‘Generalized notions of amenability’, J. Funct. Anal. 208 (2004), 229–260.CrossRefGoogle Scholar
[7]Ghahramani, F., Loy, R. J. and Zhang, Y., ‘Generalized notions of amenability, II’, J. Funct. Anal. 254 (2008), 1776–1810.CrossRefGoogle Scholar
[8]Ghahramani, F. and Zabandan, G., ‘2-Weak amenability of a Beurling algebra and amenability of its second dual’, Int. J. Pure Appl. Math. 16 (2004), 75–86.Google Scholar
[9]Ghahramani, F. and Zhang, Y., ‘Pseudo-amenable and pseudo-contractible Banach algebras’, Math. Proc. Cambridge Philos. Soc. 142 (2007), 111–123.CrossRefGoogle Scholar
[10]Grønbæk, N., ‘Amenability of weighted convolution algebras on locally compact groups’, Trans. Amer. Math. Soc. 319 (1990), 765–775.CrossRefGoogle Scholar
[11]Losert, V. and Rindler, H., ‘Asymptotically central functions and invariant extensions of Dirac measure’, in: Probability Measures on Groups VII, Lecture Notes in Mathematics, 1064 (ed. Heyer, H.) (Springer, Berlin, 1984), pp. 368–378.CrossRefGoogle Scholar
[12]Mosak, R., ‘Central functions in group algebras’, Proc. Amer. Math. Soc. 29 (1971), 613–616.CrossRefGoogle Scholar
[13]Palmer, T. W., Banach Algebras and the General Theory of *-Algebras I, II (Cambridge University Press, Cambridge, 1994, 2001).CrossRefGoogle Scholar
[14]Samei, E., ‘Weak amenability and 2-weak amenability of Beurling algebras’, J. Math. Anal. Appl. 346 (2008), 451–467.CrossRefGoogle Scholar
[15]Stokke, R., ‘Quasi-central bounded approximate identities in group algebras of locally compact groups’, Illinois J. Math. 48 (2004), 151–170.CrossRefGoogle Scholar
[16]White, M. C., ‘Characters on weighted amenable groups’, Bull. Lond. Math. Soc. 23 (1991), 375–380.CrossRefGoogle Scholar
[17]Zhang, Y., ‘Solved and unsolved problems in generalized notions of amenability for Banach algebras’, in: Banach Algebras 2009, Banach Centre Publications, 91 (eds. Loy, R. J., Sołtysiak, A. and Runde, V.) (Polish Academy of Sciences, Institute of Mathematics, Warsaw, 2010), pp. 441–454.Google Scholar