Extreme invariant means without minimal support

Fairchild, Lonnie

doi:10.1090/S0002-9947-1972-0308685-2

Extreme invariant means without minimal support
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by Lonnie Fairchild PDF

Trans. Amer. Math. Soc. 172 (1972), 83-93 Request permission

Abstract:

Let $S$ be a left amenable semigroup. We show that if $S$ has a subset satisfying a certain condition, then there is an extreme left invariant mean on $S$ whose support is not a minimal closed invariant subset of $\beta S$. Then we show that all infinite solvable groups and countably infinite locally finite groups have such subsets.

References

Ching Chou, Minimal sets and ergodic measures for $\beta N\backslash N$, Illinois J. Math. 13 (1969), 777–788. MR 249569
Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. MR 92128
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
Walter Helbig Gottschalk and Gustav Arnold Hedlund, Topological dynamics, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R.I., 1955. MR 0074810
Anthony To-ming Lau, Topological semigroups with invariant means in the convex hull of multiplicative means, Trans. Amer. Math. Soc. 148 (1970), 69–84. MR 257260, DOI 10.1090/S0002-9947-1970-0257260-5
Theodore Mitchell, Constant functions and left invariant means on semigroups, Trans. Amer. Math. Soc. 119 (1965), 244–261. MR 193523, DOI 10.1090/S0002-9947-1965-0193523-8
Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470

On amenable semigroups and applications of the Stone-Čech compactification

Carroll Wilde and Klaus Witz, Invariant means and the Stone-Čech compactification, Pacific J. Math. 21 (1967), 577–586. MR 212552