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Extreme point properties of fixed-point sets

Published online by Cambridge University Press:  17 April 2009

Rodney Nillsen
Rodney Nillsen
Affiliation:
Flinders University of South Australia, Bedford Park, South Australia The Royal University of Malta, Msida, Malta.
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Abstract

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We consider a semigroup S acting as affine continuous maps on a compact convex set X. F denotes the corresponding set of fixed points. Let exX and exF denote the corresponding sets of extreme points. If X is a simplex, conditions are given which ensure that when x ε F, the maximal measure representing x invariant under S. We also prove exF = F ∩ exX under conditions involving extreme amenability of S. Topological properties of exF are also studied.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 6 , Issue 2 , April 1972 , pp. 241 - 249
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Argabright, L.N., “Invariant means and fixed points; a sequel to Mitchell's paper”, Trans. Amer. Math. Soc. 130 (1968), 127–130.Google Scholar
[2]Bressler, D.W. and Sion, M., “The current theory of analytic sets”, Canad. J. Math. 16 (1964), 207–230.CrossRefGoogle Scholar
[3]Choquet, Gustave, Lectures on analysis, Volumes I, II, III (W.A. Benjamin, New York, Amsterdam, 1969).Google Scholar
[4]Converse, George, Namioka, Isaac and Phelps, R.R., “Extreme invariant positive operators”, Trans. Amer. Math. Soc. 137 (1969), 375–385.CrossRefGoogle Scholar
[5]Day, Mahlon M., “Fixed-point theorems for compact convex sets”, Illinois J. Math. 5 (1961), 585–590.CrossRefGoogle Scholar
[6]Dunford, Nelson and Schwartz, Jacob T., Linear operators, Part I (Interscience [John Wiley & Sons], New York, London, 1958).Google Scholar
[7]Effros, Edward G., “Structure in simplexes”, Acta Math. 117 (1967), 103–121.CrossRefGoogle Scholar
[8]Granirer, E., “Extremely amenable semigroups”, Math. Scand. 17 (1965), 177–197.CrossRefGoogle Scholar
[9]Granirer, E., “Extremely amenable semigroups II”, Math. Scand. 20 (1967), 93–113.CrossRefGoogle Scholar
[10]Halmos, Paul R., Measure theory (Van Nostrand, Toronto, New York, London, 1950).CrossRefGoogle Scholar
[11]Hewitt, Edwin and Ross, Kenneth A., Abstract harmonic analysis, Vol. I (Die Grundlehren der mathematischen Wissenschaften, Band 115. Academic Press, New York; Springer-Verlag, Berlin, Göttingen, Heidelberg, 1963).Google Scholar
[12]Jellett, Francis, “Homomorphisms and inverse limits of Choquet simplexes”, Math. Z. 103 (1968), 219–226.CrossRefGoogle Scholar
[13]Ljapin, E.S., Semigroups (Translations of Mathematical Monographs, 3. Amer. Math. Soc., Providence, Rhode Island, 1963).CrossRefGoogle Scholar
[14]Meyer, Paul A., Probability and potentials (Blaisdell, Waltham, Massachusetts; Toronto; London; 1966).Google Scholar
[15]Mitchell, Theodore, “Fixed points and multiplicative left invariant means”, Trans. Amer. Math. Soc. 122 (1966), 195–202.CrossRefGoogle Scholar
[16]Mitchell, Theodore, “Function algebras, means, and fixed points”, Trans. Amer. Math. Soc. 130 (1968), 117–126.CrossRefGoogle Scholar
[17]Phelps, Robert R., Lectures on Choquet's theorem (Van Nostrand, Princeton, New Jersey; Toronto; New York; London; 1966).Google Scholar