Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T11:35:40.694Z Has data issue: false hasContentIssue false
  • English
  • Français

The cardinality of the set of left invariant means on topological semigroups

Published online by Cambridge University Press:  17 April 2009

Heneri A.M. Dzinotyiweyi
Heneri A.M. Dzinotyiweyi
Affiliation:
University of Zimbabwe, P.O. Box MP 167, Mount Pleasant, Harare, Zimbabwe
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 very large class of topological semigroups, we establish lower and upper bounds for the cardinality of the set of left invariant means on the space of left uniformly continuous functions. In certain cases we show that such a cardinality is exactly , where b is the smallest cardinality of the covering of the underlying topological semigroup by compact sets.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 2 , April 1988 , pp. 247 - 262
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Chou, C., ‘On topologically invariant means on a locally compact group’, Trans. Amer. Math. Soc. 151 (1970), 443–456.Google Scholar
[2]Chou, C., ‘The exact cardinality of the set of invariant means on a group’, Proc. Amer. Math. Soc. 55 (1976), 103–106.CrossRefGoogle Scholar
[3]Day, M.M., ‘Lumpy subsets in left-amenable locally compact semigroups’, Pacific J. Math. 82 (1976), 87–92.Google Scholar
[4]Dzinotyiweyi, H.A.M., ‘Algebras of measures on C-distinguished topological semigroups’, Math. Proc. Cambridge Philos. Soc. 84 (1978), 323–336.CrossRefGoogle Scholar
[5]Dzinotyiweyi, H.A.M., The analogue of the group algebra for topological semigroups (Pitman Books Ltd., Boston, London, Melbourne, 1984).Google Scholar
[6]Dzinotyiweyi, H.A.M., ‘Nonseparability of quotient spaces of function algebras on topological semigroups’, Thans. Amer. Math. Soc. 272, 223–235.CrossRefGoogle Scholar
[7]Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, New York, 1960).CrossRefGoogle Scholar
[8]Hewitt, E. and Ross, K.A., Abstract harmonic analysis Vol. 1 (Springer-Verlag, Berlin, 1963).Google Scholar
[9]Klawe, M.M., ‘Dimensions of the sets of invariant means of semigroups’, Illinois J. Math. 24 (1980), 233–243.CrossRefGoogle Scholar
[10]Lau, A.T.M. and Paterson, A.L.T., ‘The exact cardinality of the set of topological left invariant means on an amenable locally compact group’, Proc. Amer. Math. Soc. 98 (1986), 75–80.CrossRefGoogle Scholar
[11]Liu, T.S. and Van Rooij, A., ‘Invariant means on a locally compact group’, Monatsh. Math. 78 (1974), 356–359.CrossRefGoogle Scholar
[12]Rosenblatt, J.M., ‘The number of extensions of an invariant mean’, Composito Math. 33 (1976), 147–159.Google Scholar
[13]Sleijpen, G.L.G., ‘Locally compact semigroups and continuous translations of measures’, Proc. London Math. Soc. 37 (1978), 75–97.Google Scholar
[14]Sleijpen, G.L.G., ‘A note on measures vanishing on emaciated sets’, Bull. London Math. Soc. 13 (1981), 219–223.Google Scholar