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Order structure on certain classes of ideals in group algebras and amenability

Published online by Cambridge University Press:  17 April 2009

Yuji Takahashi
Affiliation:
Department of Mathematics, Hokkaido University of Education, Hakodate 040–8567, Japan, e-mail: [email protected],ac.jp
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Abstract

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Let G be a separable, locally compact group and let d (G) be the set of all closed left ideals in L1(G) which have the form Jμ = {ff ∗ μ: fL1(G)} for some discrete probability measure μ. It is shown that if d (G) has a unique maximal element with respect to the order structure by set inclusion, then G is amenable. This answers a problem of G.A. Willis. We also examine cardinal numbers of the sets of maximal elements in d (G) for nonamenable groups.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Hewitt, E. and Ross, K.A., Abstract harmonic analysis I, 2nd Edition (Springer-Verlag, Berlin, Heidelberg, New York, 1979).CrossRefGoogle Scholar
[2]Kakutani, S. and Kodaira, K., ‘Über das Haarsche Mass in der lokal bikompakten. Gruppe’, Proc. Imp. Acad. Tokyo 20 (1944), 444450.Google Scholar
[3]Paterson, A.L.T., Amenability, Math. Surveys and Monographs 29 (Amer. Math. Soc., Providence, RI, 1988).CrossRefGoogle Scholar
[4]Pier, J.-P., Amenable locally compact groups (John Wiley and Sons, New York, 1984).Google Scholar
[5]Reiter, H., ‘Sur certains idéaux dans L 1(G)’, C.R. Acad. Sci. Paris Sér. A - B 267 (1968), 882885.Google Scholar
[6]Reiter, H., L1-Algebras and Segal algebras, Lecture Notes in Mathematics 231 (Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
[7]Reiter, H. and Stegeman, J.D., Classical harmonic analysis and locally compact groups, London Math. Soc. Monographs 22 (Clarendon Press, Oxford, 2000).CrossRefGoogle Scholar
[8]Rosenblatt, J., ‘Ergodic and mixing random walks on locally compact groups’, Math. Ann. 257 (1981), 3142.CrossRefGoogle Scholar
[9]Willis, G.A., ‘Probability measures on groups and some related ideals in group algebras’, J. Functional Analysis 92 (1990), 202263.CrossRefGoogle Scholar