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On the equivalence of invariant integrals and minimal ideals in semigroups

Published online by Cambridge University Press:  17 April 2009

N. A. Tserpes  and
A. G. Kartsatos
N. A. Tserpes
Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida, USA.
A. G. Kartsatos
Affiliation:
Department of Mathematics, University of Athens, Athens, Greece.
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Abstract

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Let S be a Hausdorff topological semigroup and Cb,(S), Cc (S), the spaces of real valued continuous functions on S which are respectively bounded and have compact support. A regular measure m on S is r*-invarient if m(B) = for every Borel B ⊂ S and every x ∈ S, where tx: s → sx is the right translation by x. The following theorem is proved: Let S be locally compact metric with the tx's closed. Then the following statements are equivalent: (i) S admits a right invariant integral on Cc (S). (ii) S admits an r*–invariant measure, (iii) S has a unique minimal left ideal. The above equivalence is considered also for normal semigroups and analogous results are obtained for finitely additive r*–invariant measures. Also in the case when S is a complete separable metric semigroup with the tx's closed, the following statements are equivalent: (i) S admits a right invariant integral I on Cb(S) such that I(1) = 1 and satisfying Daniel's condition. (ii) S admits an r*–invariant probability measure. (iii) S has a right ideal which is a compact group and which is contained in a unique minimal left ideal. Finally, in order that a locally compact S admit a right invariant measure, it suffices that S contain a right ideal F which is a left group such that (B ∩ F)x = BX ∩ Fx for all Borel B ⊂ S.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 1 , Issue 2 , August 1969 , pp. 269 - 278
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Argabright, L., “Invariant means on topological semigroups”, Pacific J. Math. 16 (1966), 193–203.CrossRefGoogle Scholar
[2]Argabright, L., “A note on invariant integrals on locally compact semigroups”, Proc. Amer. Math. Soc. 17 (1966), 377–382.CrossRefGoogle Scholar
[3]Berberian, S.K., Measure and Integration, (MacMillan Co., New York, 1965).Google Scholar
[3a]Bourbaki, Nicolas, General Topology, Part 1, (Addison-Wesley, Reading, Mento Park, London, Ontario, 1966).Google Scholar
[3b]Gillman, Leonard and Jerison, Meyer, Rings of continuous functions, (Van Nostrand, Princeton, 1960).CrossRefGoogle Scholar
[4]Heble, M. and Rosenblatt, M., “Idempotent measures on a compact topological semigroup”, Proc. Amer. Math. Soc. 14 (1963), 177–184.CrossRefGoogle Scholar
[5]Hewitt, Edwin, “Linear functionals on spaces of continuous functions”, Fund. Math. 37 (1950), 161–189.CrossRefGoogle Scholar
[6]Hewitt, Edwin and Ross, Kenneth A., Abstract harmonic analysis (Academic Press, New York, London, 1963).Google Scholar
[7]Markov, A., “On mean values and exterior densities”, Rec. math. Moscou (Mat. Sb.) 4 (1938), 165–190.Google Scholar
[8]Michael, J.H., “Right invariant integrals on locally compact semigroups”, J. Austral. Math. Soc. 4 (1964), 273–286.CrossRefGoogle Scholar
[9]Morita, Kiiti and Hanai, Satirọ, “Closed mappings and metric spaces”, Proc. Japan Acad. 32 (1956), 10–14.Google Scholar
[10]Neveu, J., Mathematical foundations of the calculus of probabilities (Holden-Day Inc., San Francisco, 1965).Google Scholar
[11]Oxtoby, John C., “Invariant measures in groups which are not locally compact”, Trans. Amer. Math. Soc. 60 (1946), 215–237.CrossRefGoogle Scholar
[12]Parthasarathy, K.R., Probability measures on metric spaces (Academic Press, New York, London, 1967).CrossRefGoogle Scholar
[13]Preston, G.B., “Inverse semigroups with minimal right ideals”, J. London Math. Soc. 29 (1954), 411–419.CrossRefGoogle Scholar