On topologically invariant means on a locally compact group

Chou, Ching

doi:10.1090/S0002-9947-1970-0269780-8

On topologically invariant means on a locally compact group
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by Ching Chou

Trans. Amer. Math. Soc. 151 (1970), 443-456

DOI: https://doi.org/10.1090/S0002-9947-1970-0269780-8

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Abstract:

Let $\mathcal {M}$ be the set of all probability measures on $\beta N$. Let G be a locally compact, noncompact, amenable group. Then there is a one-one affine mapping of $\mathcal {M}$ into the set of all left invariant means on ${L^\infty }(G)$. Note that $\mathcal {M}$ is a very big set. If we further assume G to be $\sigma$-compact, then we have a better result: The set $\mathcal {M}$ can be embedded affinely into the set of two-sided topologically invariant means on ${L^\infty }(G)$. We also give a structure theorem for the set of all topologically left invariant means when G is $\sigma$-compact.

References

Ching Chou, Minimal sets and ergodic measures for $\beta N\backslash N$, Illinois J. Math. 13 (1969), 777–788. MR 249569
Ching Chou, On the size of the set of left invariant means on a semi-group, Proc. Amer. Math. Soc. 23 (1969), 199–205. MR 247444, DOI 10.1090/S0002-9939-1969-0247444-1
Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. MR 92128
Mahlon M. Day, Normed linear spaces, Reihe: Reelle Funktionen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0094675
W. R. Emerson and F. P. Greenleaf, Covering properties and Følner conditions for locally compact groups, Math. Z. 102 (1967), 370–384. MR 220860, DOI 10.1007/BF01111075
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
E. Granirer, On amenable semigroups with a finite-dimensional set of invariant means. I, Illinois J. Math. 7 (1963), 32–48. MR 144197
Edmond Granirer, On the invariant mean on topological semigroups and on topological groups, Pacific J. Math. 15 (1965), 107–140. MR 209388
E. Granirer, On Baire measures on $D$-topological spaces, Fund. Math. 60 (1967), 1–22. MR 208355, DOI 10.4064/fm-60-1-1-22
Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto-London, 1969. MR 0251549
A. Hulanicki, Means and Følner condition on locally compact groups, Studia Math. 27 (1966), 87–104. MR 195982, DOI 10.4064/sm-27-2-87-104
J. M. Kister, Uniform continuity and compactness in topological groups, Proc. Amer. Math. Soc. 13 (1962), 37–40. MR 133392, DOI 10.1090/S0002-9939-1962-0133392-8
Indar S. Luthar, Uniqueness of the invariant mean on Abelian topological semigroups, Trans. Amer. Math. Soc. 104 (1962), 403–411. MR 150232, DOI 10.1090/S0002-9947-1962-0150232-6
I. Namioka, Følner’s conditions for amenable semi-groups, Math. Scand. 15 (1964), 18–28. MR 180832, DOI 10.7146/math.scand.a-10723
Ralph A. Raimi, On Banach’s generalized limits, Duke Math. J. 26 (1959), 17–28. MR 117569
Neil W. Rickert, Amenable groups and groups with the fixed point property, Trans. Amer. Math. Soc. 127 (1967), 221–232. MR 222208, DOI 10.1090/S0002-9947-1967-0222208-6