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

  • Yuji Takahashi (a1)

Abstract

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.

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Copyright

References

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[1]Hewitt, E. and Ross, K.A., Abstract harmonic analysis I, 2nd Edition (Springer-Verlag, Berlin, Heidelberg, New York, 1979).
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[6]Reiter, H., L1-Algebras and Segal algebras, Lecture Notes in Mathematics 231 (Springer-Verlag, Berlin, Heidelberg, New York, 1971).
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[8]Rosenblatt, J., ‘Ergodic and mixing random walks on locally compact groups’, Math. Ann. 257 (1981), 3142.
[9]Willis, G.A., ‘Probability measures on groups and some related ideals in group algebras’, J. Functional Analysis 92 (1990), 202263.
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Order structure on certain classes of ideals in group algebras and amenability

  • Yuji Takahashi (a1)

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