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Translation invariant functionals on Lp (G) when G is not amenable

Part of: Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

G. A. Willis
G. A. Willis
Affiliation:
Department of Pure Mathematics, The University of Adelaide, Box 498 G.P.O., Adelaide S.A. 5001, Australia
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Abstract

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It is shown that if G is a non-amenable group, then there are no non-zero translation invariant functionals on Lp(G) for 1 < p < ∞. Furthermore, if G contains a closed, non-abelian free subgroup, then there are no non-zero translation invariant functionals on C0(G). The latter is proved by showing that a certain non-invertible convolution operator on C0(G) is surjective.

MSC classification

Secondary: 43A07: Means on groups, semigroups, etc.; amenable groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 2 , October 1986 , pp. 237 - 250
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Day, M. M., ‘Convolution, means and spectra’, Illinois J. Math. 8 (1964), 100–111.CrossRefGoogle Scholar
[2]Johnson, B. E., ‘Continuity of homomorphisms of Banach G-modules’, preprint.Google Scholar
[3]Johnson, B. E., ‘Some examples in harmonic analysis’, Studia Math. 48 (1973), 181–188.CrossRefGoogle Scholar
[4]Ludvik, P., ‘Discontinuous translation-invariant linear functionals on L 1(G)’, Studia Math. 56 (1976), 21–30.CrossRefGoogle Scholar
[5]Lyndon, R. C. and Schupp, P. E., Combinatorial group theory (Ergeb. der Math., Vol. 89, Springer-Verlag, Berlin and New York, 1977).Google Scholar
[6]Meisters, G. H., ‘Some problems and results on translation-invariant linear forms’ (Proc. of Conference on Radical Banach Algebras and Automatic Continuity, Long Beach, 1981, Lecture Notes in Math., Vol. 975, Springer-Verlag, Berlin and New York, 1983, pp. 423–444).Google Scholar
[7]Reiter, H., Classical harmonic analysis and locally compact groups (Oxford University Press, 1968).Google Scholar
[8]Reiter, H., L1-algebras and Segal algebras (Lecture Notes in Math., Vol. 231, Springer-Verlag, Berlin and New York, 1971).CrossRefGoogle Scholar
[9]Reiter, H., ‘Sur certaines idéaux dans L 1(G)’, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), 882–885.Google Scholar
[10]Saeki, S., ‘Discontinuous translation invariant functionals’, Trans. Amer. Math. Soc. 282 (1984), 403–414.CrossRefGoogle Scholar
[11]Sinclair, A. M., Automatic continuity of linear operators (London Math. Soc. Lecture Note Series 21, Cambridge University Press, 1976).CrossRefGoogle Scholar
[12]Woodward, G. S., ‘Translation-invariant linear forms on C 0(G), C(G), L P(G) for non-compact groups’, J. Funct. Anal. 16 (1974), 205–220.CrossRefGoogle Scholar