Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T16:38:01.621Z Has data issue: false hasContentIssue false
  • English
  • Français

Derivations from Totally Ordered Semigroup Algebras into their Duals

Published online by Cambridge University Press:  20 November 2018

T. D. Blackmore
T. D. Blackmore*
Affiliation:
Department of Pure Mathematics University of Sheffield Sheffield England S3 7RH
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.

For a well-behaved measure μ, on a locally compact totally ordered set X, with continuous part μc, we make Lp (X, μc) into a commutative Banach bimodule over the totally ordered semigroup algebra Lp (X, μ), in such a way that the natural surjection from the algebra to the module is a bounded derivation. This gives rise to bounded derivations from Lp (X, μ) into its dual module and in particular shows that if μc is not identically zero then Lp (X, μ) is not weakly amenable. We show that all bounded derivations from L1 (X, μ) into its dual module arise in this way and also describe all bounded derivations from Lp(X, μ) into its dual for 1 < p < ∞ the case that X is compact and μ continuous.

Keywords

43A2046M20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 2 , 01 June 1997 , pp. 133 - 142
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Bade, W. G., Curtis, P. C. and Dales, H. G., Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. 55 (1987), 359–377.Google Scholar
2. Baker, J. W., Pym, J. S. and Vasudeva, H. L., Totally ordered measure spaces and their Lp algebras, Mathematika 29 (1982), 42–54.Google Scholar
3. Blackmore, T. D., Weak amenability of discrete convolution algebras, Semigroup Forum, to appear.Google Scholar
4. Groenbæk, N., A characterisation of weak amenability, Studia Math. 94 (1987), 149–162.Google Scholar
5. Groenbæk, N., Commutative Banach algebras, module derivations and semigroups, J. London Math. Soc. 40 (1989), 137–157.Google Scholar
6. Helemskii, A. Ya., Banach and Locally Convex Algebras, Oxford University Press, 1993.Google Scholar
7. Hewitt, E. and Zuckerman, H. S., Structure theory for a class of convolution algebras, Pacific J. Math. 7 (1957), 913–941.Google Scholar
8. Johnson, B. E., Weak amenability of group algebras, Bull. London Math. Soc. 23 (1991), 281–284.Google Scholar
9. Kelley, J. L., General Topology, D. Van Nostrand Co. Inc., 1955.Google Scholar
10. Lau, A. T-M., Amenability of semigroups. In: The Analytical and Topological Theory of Semigroups, (ed. Hoffmann, K. H., et al.,), de Gruyter, Berlin, 1990, 313–334.Google Scholar
11. Ross, K. A., The structure of certain measure algebras, Pacific J. Math. 11 (1961), 723–737.Google Scholar