The exact cardinality of the set of topological left invariant means on an amenable locally compact group

Authors:
Anthony To Ming Lau and Alan L. T. Paterson

Journal:
Proc. Amer. Math. Soc. **98** (1986), 75-80

MSC:
Primary 43A07; Secondary 43A15

MathSciNet review:
848879

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Abstract: The purpose of this note is to prove that if is an amenable locally compact noncompact group, then the set of topological left invariant means on has cardinality , where is the smallest cardinality of the covering of by compact sets. We also prove that in this case the spectrum of the bounded left uniformly continuous complex-valued functions contains exactly minimal closed invariant subsets (or left ideals)

**[1]**J. W. Baker and P. Milnes,*The ideal structure of the Stone-Čech compactification of a group*, Math. Proc. Cambridge Philos. Soc.**82**(1977), no. 3, 401–409. MR**0460516****[2]**Ching Chou,*On topologically invariant means on a locally compact group*, Trans. Amer. Math. Soc.**151**(1970), 443–456. MR**0269780**, 10.1090/S0002-9947-1970-0269780-8**[3]**Ching Chou,*The exact cardinality of the set of invariant means on a group*, Proc. Amer. Math. Soc.**55**(1976), no. 1, 103–106. MR**0394036**, 10.1090/S0002-9939-1976-0394036-3**[4]**Mahlon M. Day,*Fixed-point theorems for compact convex sets*, Illinois J. Math.**5**(1961), 585–590. MR**0138100****[5]**Mahlon Marsh Day,*Correction to my paper “Fixed-point theorems for compact convex sets”*, Illinois J. Math.**8**(1964), 713. MR**0169210****[6]**Edmond E. Granirer,*Exposed points of convex sets and weak sequential convergence*, American Mathematical Society, Providence, R.I., 1972. Applications to invariant means, to existence of invariant measures for a semigroup of Markov operators etc. . ; Memoirs of the American Mathematical Society, No. 123. MR**0365090****[7]**Frederick P. Greenleaf,*Invariant means on topological groups and their applications*, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR**0251549****[8]**E. Hewitt and K. Ross,*Abstract harmonic analysis*1, Springer-Verlag, Berlin and New York, 1963.**[9]**Anthony To Ming Lau,*Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups*, Math. Proc. Cambridge Philos. Soc.**99**(1986), no. 2, 273–283. MR**817669**, 10.1017/S0305004100064197**[10]**Theodore Mitchell,*Constant functions and left invariant means on semigroups*, Trans. Amer. Math. Soc.**119**(1965), 244–261. MR**0193523**, 10.1090/S0002-9947-1965-0193523-8**[11]**Teng Sun Liu and Arnoud van Rooij,*Invariant means on a locally compact group*, Monatsh. Math.**78**(1974), 356–359. MR**0358218****[12]**Alan L. T. Paterson,*The cardinality of the set of left invariant means on a left amenable semigroup*, Illinois J. Math.**29**(1985), no. 4, 567–583. MR**806467****[13]**Jean-Paul Pier,*Amenable locally compact groups*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR**767264****[14]**Joseph Max Rosenblatt,*The number of extensions of an invariant mean*, Compositio Math.**33**(1976), no. 2, 147–159. MR**0435729**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1986-0848879-6

Keywords:
Amenable locally compact groups,
invariant means,
left thick subsets,
uniformly continuous functions,
minimal invariant sets

Article copyright:
© Copyright 1986
American Mathematical Society