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The continuity of derivations from group algebras: factorizable and connected groups

Part of: Abstract harmonic analysis Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  09 April 2009

George Willis
George Willis
Affiliation:
Mathematics Research SectionSchool of Mathematical Sciences Australian National UniversityGPO Box 4, Canberra ACT 2601, Australia
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Abstract

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A group is said to be factorizable if it has a finite number of abelian subgroups, H1, H2, … Hn, such that G = H1H2 … Hn. It is shown that, if G is a factorizable or connected locally compact group, then every derivation from L1 (G) to an arbitrary L1 (G)-bimodule X is continuous.

Keywords

factorizationgroup algebraautomatic continuityderivation.

MSC classification

Secondary: 46H05: General theory of topological algebras 46H10: Ideals and subalgebras 43A20: $L^1$-algebras on groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 52 , Issue 2 , April 1992 , pp. 185 - 204
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Bade, W. G. & Curtis, P. C. Jr, ‘The continuity of derivations of Banach algebras’, J. Funct. Anal. 16 (1974), 372–387.CrossRefGoogle Scholar
[2]Bonsall, F. F. & Duncan, J.Complete normed algebras (Springer-Verlag, Berlin-Heidelberg-New York, 1973).CrossRefGoogle Scholar
[3]Dales, H. G., ‘Automatic continuity: a survey’, Bull. London Math. Soc. 10 (1978), 129–183.CrossRefGoogle Scholar
[4]Dales, H. G. & Willis, G. A. ‘Cofinite ideals in Banach algebras and finite-dimensional representations of group algebras’, Automatic continuity and radical Banach algebras (Lecture Notes in Math., vol. 975, Springer-Verlag, Berlin-Heidelberg-New York, 1982).Google Scholar
[5]Derighetti, A., ‘Some remarks on L1(G)’, Math. Z. 164 (1978), 189–194.CrossRefGoogle Scholar
[6]Dunkl, C. F. & Ramirez, D. E., Topics in harmonic analysis (Appleton-Century-Crofts, New York, 1971).Google Scholar
[7]Hewitt, E. & Ross, K. A.Abstract harmonic analysis, Springer-Verlag, Berlin-Heidelberg-New York, 1979.CrossRefGoogle Scholar
[8]Iwasawa, K., ‘On some types of topological groups’, Annals of Maths. 50 (1949), 507–558.CrossRefGoogle Scholar
[9]Jewel, N. P. ‘Continuity of module and higher derivations’ Pacific J. Math. 68 (1977), 91–98.CrossRefGoogle Scholar
[10]Johnson, B. E. & Parrott, S. K., ‘Operators commuting with a von Neumann algebra modulo the set of compact operators’, J. of Functional Analysis 11 (1972), 39–61.CrossRefGoogle Scholar
[11]Johnson, B. E. & Sinclair, A. M., ‘Continuity of derivations and a problem of Kaplansky’, Amer. J. Math. 94 (1972), 685–698.Google Scholar
[12]Liu, T.-S., van Rooij, A. & Wang, J.-K., ‘Projections and approximate identities for ideals in group algebras’, Trans. Amer. Math. Soc. 175 (1973), 181–188.CrossRefGoogle Scholar
[13]Montgomery, D. & Zippin, L., Topological transformation groups (Interscience, New York, 1955).Google Scholar
[14]Plaumann, P. ‘Polythetische Gruppen und ihre Automorphismengruppen’ Mh. Math. 100 (1985), 145–151.Google Scholar
[15]Reiter, H., Classical harmonic analysis and locally compact groups (Oxford University Press, 1968).Google Scholar
[16]Ringrose, J. R., ‘Automatic continuity of derivations of operator algebras’ J. London Math. Soc. (2) 5 (1972), 432–438.CrossRefGoogle Scholar
[17]Rudin, W., Fourier analysis on groups (Interscience, New York, 1962).Google Scholar
[18]Sinclair, A. M., Automatic continuity of linear operators (Cambridge University Press, Cambridge, 1976).CrossRefGoogle Scholar
[19]Willis, G. A., ‘Approximate units in finite codimensional ideals of group algebras’ J. London Math. Soc. (2) 26 (1982), 143–154.Google Scholar
[20]Willis, G. A., ‘The continuity of derivations and module homomorphisms J. Austral. Math. Soc. (Series A) 40 (1986), 299–320.CrossRefGoogle Scholar
[21]Willis, G. A., ‘The continuity of left l 1 (G)-module homomorphisms from L1 (G)’ (to appear).Google Scholar
[22]Willis, G. A., ‘Probability measures on groups and related ideals in group algebras’ J. of Functional Analysis 92 (1990), 202–263.CrossRefGoogle Scholar