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Uniqueness of invariant means on certain introverted spaces

Published online by Cambridge University Press:  17 April 2009

Marvin W. Grossman
Marvin W. Grossman
Affiliation:
Department of Mathematics, Temple University, Philadelphia, Pennsylvania, USA.
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Abstract

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Let S be a topological semigroup with separately continuous multiplication and H a uniformly closed invariant subspace of LUC(S) (the space of left uniformly continuous bounded functions on S ) that contains the constants. It is shown that if H is left introverted and H admits a tight two-sided invariant mean m, then for each h ∈ H, m(h) is the unique constant function in the norm closed convex hull of the left orbit of h; consequently, H has a unique left invariant mean. (In fact, it is enough for H to admit a tight right invariant mean and a left invariant mean. ) For certain S, a similar result is obtained when H is a left compact-open introverted subspace of LCC(S) (the space of left compact-open continuous functions on S ).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 9 , Issue 1 , August 1973 , pp. 109 - 120
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Argabright, L., “Invariant means on topological semigroups”, Pacific J. Math. 16 (1966), 193–203.CrossRefGoogle Scholar
[2]Argabright, L.N., “Invariant means and fixed points; a sequel to Mitchell's paper”, Trans. Amer. Math. Soc. 130 (1968), 127–130.Google Scholar
[3]Burckel, R.B., Weakly almost periodic functions on semigroups (Gordon and Breach, New York, London, Paris, 1970).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]Gillman, Leonard and Jerison, Meyer, Rings of continuous functions (Van Hostrand, Princeton, New Jersey; Toronto; London; New York; 1960).Google Scholar
[6]Gould, G.G. and Mahowald, M., “Measures on completely regular spaces”, J. London Math. Soc. 37 (1962), 103–111.CrossRefGoogle Scholar
[7]Granirer, E., “On amenable semigroups with a finite-dimensional set of invariant means I”, Illinois J. Math. 7 (1963), 32–48.Google Scholar
[8]Granirer, E., “On amenable semigroups with a finite-dimensional set of invariant means II”, Illinois J. Math. 7 (1963), 49–58.Google Scholar
[9]Granirer, E. and Lau, Anthony T., “Invariant means on locally compact groups”, Illinois J. Math. 15 (1971), 249–257.CrossRefGoogle Scholar
[10]Grossman, Marvin W., “A categorical approach to invariant means and fixed point properties”, Semigroup Forum 5 (1972), 14–44CrossRefGoogle Scholar
[11]Halmos, Paul R., Measure theory (Van Nostrand, Toronto, New York, London, 1950).CrossRefGoogle Scholar
[12]Kelley, J.L., Namioka, Isaac and Donoghue, W.F. Jr, Lucas, Kenneth R., Pettis, B.J., Poulsen, Ebbe Thue, Price, G. Baley, Robertson, Wendy, Scott, W.R., Smith, Kennan T., Linear topological spaces (Van Hostrand, Princeton, New Jersey; Toronto; New York; London; 1963).CrossRefGoogle Scholar
[13]Khurana, Surjit Singh, “Measures and barycenters of measures on convex sets in locally convex spaces I”, J. Math. Anal. Appl. 27 (1969), 103–115.CrossRefGoogle Scholar
[14]Khurana, Surjit Singh, “Measures and barycenters of measures on convex sets in locally convex spaces II”, J. Math. Anal. Appl. 28 (1969), 222–229.Google Scholar
[15]Lau, Anthony To-ming, “Action of topological semigroups, invariant means, and fixed points”, Studia Math. 43 (1972), 139–156.Google Scholar
[16]Mitchell, Theodore, “Constant functions and left invariant means on semigroups”, Trans. Amer. Math. Soc. 119 (1965), 244–261.CrossRefGoogle Scholar
[17]Mitchell, Theodore, “Function algebras, means, and fixed points”, Trans. Amer. Math. Soc. 130 (1968), 117–126.CrossRefGoogle Scholar
[18]Peressini, Anthony L., Ordered topological vector spaces (Harper and Row, New York, Evanston, London, 1967).Google Scholar
[19]Птак, Власимил [Vlastimil Pták], “О голных тогологических линейных гространствах” [On complete topological linear spaces], Čehoslovack. Mat. Ž. 3 (78) (1953), 301–364.Google ScholarPubMed