Topological semigroups and representations. I

Wong, James C. S.

doi:10.1090/S0002-9947-1974-0369604-8

Topological semigroups and representations. I
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by James C. S. Wong

Trans. Amer. Math. Soc. 200 (1974), 89-109

DOI: https://doi.org/10.1090/S0002-9947-1974-0369604-8

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Abstract:

Let $S$ be a topological semigroup (separately continuous multiplication) with identity and $W(S)$ the Banach space of all weakly almost periodic functions on $S$. It is well known that if $S = G$ is a locally compact group, then $W(G)$ always has a (unique) invariant mean. In other words, there exists $m \in W{(G)^ \ast }$ such that $||m|| = m(1) = 1$ and $m({l_s}f) = m({r_s}f) = m(f)$ for any $s \in G,f \in W(G)$ where ${l_s}f(t) = f(st)$ and ${r_s}f(t) = f(ts),t \in S$ The main purpose of this paper is to present several characterisations (functional analytic and algebraic) of the existence of a left (right) invariant mean on $W(S)$ In particular, we prove that $W(S)$ has a left (right) invariant mean iff a certain compact topological semigroup $p{(S)^\omega }$ (to be defined) associated with $S$ contains a right (left) zero. Other results in this direction are also obtained.

References

L. N. Argabright, Invariant means and fixed points: A sequel to Mitchell’s paper, Trans. Amer. Math. Soc. 130 (1968), 127–130. MR 217578, DOI 10.1090/S0002-9947-1968-0217578-X
Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. MR 92128
K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63–97. MR 131784, DOI 10.1007/BF02559535
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217–240. MR 36455, DOI 10.1090/S0002-9947-1949-0036455-9
E. Granirer, Extremely amenable semigroups, Math. Scand. 17 (1965), 177–197. MR 197595, DOI 10.7146/math.scand.a-10772
E. Granirer, Extremely amenable semigroups. II, Math. Scand. 20 (1967), 93–113. MR 212551, DOI 10.7146/math.scand.a-10825
Edmond E. Granirer, Functional analytic properties of extremely amenable semigroups, Trans. Amer. Math. Soc. 137 (1969), 53–75. MR 239408, DOI 10.1090/S0002-9947-1969-0239408-3
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
R. D. Holmes and Anthony T. Lau, Non-expansive actions of topological semigroups and fixed points, J. London Math. Soc. (2) 5 (1972), 330–336. MR 313895, DOI 10.1112/jlms/s2-5.2.330
John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
Anthomy ToMing Lau, Invariant means on almost periodic functions and fixed point properties, Rocky Mountain J. Math. 3 (1973), 69–76. MR 324313, DOI 10.1216/RMJ-1973-3-1-69
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
Theodore Mitchell, Fixed points and multiplicative left invariant means, Trans. Amer. Math. Soc. 122 (1966), 195–202. MR 190249, DOI 10.1090/S0002-9947-1966-0190249-2
Theodore Mitchell, Function algebras, means, and fixed points, Trans. Amer. Math. Soc. 130 (1968), 117–126. MR 217577, DOI 10.1090/S0002-9947-1968-0217577-8
Theodore Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630–641. MR 270356
Theodore Mitchell, Fixed points of reversible semigroups of nonexpansive mappings, K\B{o}dai Math. Sem. Rep. 22 (1970), 322–323. MR 267414
Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470