On a geometric property of the set of invariant means on a group
Author:
Ching Chou
Journal:
Proc. Amer. Math. Soc. 30 (1971), 296302
MSC:
Primary 46.80; Secondary 42.00
MathSciNet review:
0283584
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Abstract: If G is a discrete group and then denotes the homeomorphism of onto induced by left multiplication by x. A subset K of is said to be invariant if it is closed, nonempty and for each . Let denote the set of left invariant means on G. (They can be considered as measures on .) Let G be a countably infinite amenable group and let K be an invariant subset of . Then the nonempty compact convex set has no exposed points (with respect to topology). Therefore, it is infinite dimensional.
 [1]
A.
P. Calderon, A general ergodic theorem, Ann. of Math. (2)
58 (1953), 182–191. MR 0055415
(14,1071a)
 [2]
Ching
Chou, On the size of the set of left
invariant means on a semigroup, Proc. Amer.
Math. Soc. 23
(1969), 199–205. MR 0247444
(40 #710), http://dx.doi.org/10.1090/S00029939196902474441
 [3]
Ching
Chou, On a conjecture of E. Granirer
concerning the range of an invariant mean, Proc. Amer. Math. Soc. 26 (1970), 105–107. MR 0260899
(41 #5519), http://dx.doi.org/10.1090/S0002993919700260899X
 [4]
Ching
Chou, On topologically invariant means on a
locally compact group, Trans. Amer. Math.
Soc. 151 (1970),
443–456. MR 0269780
(42 #4675), http://dx.doi.org/10.1090/S00029947197002697808
 [5]
Mahlon
M. Day, Amenable semigroups, Illinois J. Math.
1 (1957), 509–544. MR 0092128
(19,1067c)
 [6]
Mahlon
M. Day, Fixedpoint theorems for compact convex sets, Illinois
J. Math. 5 (1961), 585–590. MR 0138100
(25 #1547)
 [7]
L. R. Fairchild, Extreme invariant means and minimal sets in the StoneČech compactification of a semigroup, Thesis, University of Illinois, Urbana, I11., 1970.
 [8]
E.
Granirer, On amenable semigroups with a finitedimensional set of
invariant means. I, Illinois J. Math. 7 (1963),
32–48. MR
0144197 (26 #1744)
 [9]
Meyer
Jerison, The set of all generalized limits of bounded
sequences, Canad. J. Math. 9 (1957), 79–89. MR 0083697
(18,747g)
 [10]
V.
L. Klee Jr., Extremal structure of convex sets. II, Math. Z.
69 (1958), 90–104. MR 0092113
(19,1065b)
 [11]
I.
Namioka, Følner’s conditions for amenable
semigroups, Math. Scand. 15 (1964), 18–28. MR 0180832
(31 #5062)
 [12]
Ralph
A. Raimi, Minimal sets and ergodic measures in
𝛽𝑁𝑁, Bull. Amer. Math.
Soc. 70 (1964),
711–712. MR 0166331
(29 #3608), http://dx.doi.org/10.1090/S000299041964111800
 [13]
Walter
Rudin, Averages of continuous functions on compact spaces,
Duke Math. J. 25 (1958), 197–204. MR 0098313
(20 #4774)
 [14]
A.
A. Tempel′man, Ergodic theorems for general dynamical
systems, Dokl. Akad. Nauk SSSR 176 (1967),
790–793 (Russian). MR 0219700
(36 #2779)
 [15]
Carroll
Wilde and Klaus
Witz, Invariant means and the StoneČech
compactification, Pacific J. Math. 21 (1967),
577–586. MR 0212552
(35 #3423)
 [1]
 A. P. Calderón, A general ergodic theorem, Ann. of Math. (2) 58 (1953), 182191. MR 14, 1071. MR 0055415 (14:1071a)
 [2]
 C. Chou, On the size of the set of left invariant means on a semigroup, Proc. Amer. Math. Soc. 23 (1969), 199205. MR 40 #710. MR 0247444 (40:710)
 [3]
 , On a conjecture of E. Granirer concerning the range of an invariant mean, Proc. Amer. Math. Soc. 26 (1970), 105107. MR 0260899 (41:5519)
 [4]
 , On topologically invariant means on a locally compact group, Trans. Amer. Math. Soc. 151 (1970), 443456. MR 0269780 (42:4675)
 [5]
 M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509544. MR 19, 1067. MR 0092128 (19:1067c)
 [6]
 , Fixedpoint theorems for compact convex sets, Illinois J. Math. 5 (1961), 585590. MR 25 #1547. MR 0138100 (25:1547)
 [7]
 L. R. Fairchild, Extreme invariant means and minimal sets in the StoneČech compactification of a semigroup, Thesis, University of Illinois, Urbana, I11., 1970.
 [8]
 E. Granirer, On amenable semigroups with a finitedimensional set of invariant means. I, II, Illinois J. Math. 7 (1963), 3258. MR 26 #1744; 1745. MR 0144197 (26:1744)
 [9]
 M. Jenison, The set of all generalized limits of bounded sequences, Canad. J. Math. 9 (1957), 7989. MR 0083697 (18:747g)
 [10]
 V. L. Klee, Jr., Extremal structure of convex sets. II, Math. Z. 69 (1958), 90104. MR 19, 1065. MR 0092113 (19:1065b)
 [11]
 I. Namioka, Følner's conditions for amenable semigroups, Math. Scand. 15 (1964), 1828. MR 31 #5062. MR 0180832 (31:5062)
 [12]
 R. A. Raimi, Minimal sets and ergodic measures in , Bull. Amer. Math. Soc. 70 (1964), 711712. MR 29 #3608. MR 0166331 (29:3608)
 [13]
 W. Rudin, Averages of continuous functions on compact spaces, Duke Math. J. 25 (1958), 197204. MR 20 #4774. MR 0098313 (20:4774)
 [14]
 A. A. Tempel'man, Ergodic theorems for general dynamic systems, Dokl. Akad. Nauk SSSR 176 (1967), 790793 = Soviet Math. Dokl. 8 (1967), 12131216. MR 36 #2779. MR 0219700 (36:2779)
 [15]
 C. Wilde and K. Witz, Invariant means and the StoneČech compactification, Pacific J. Math. 21 (1967), 577586. MR 35 #3423. MR 0212552 (35:3423)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197102835848
PII:
S 00029939(1971)02835848
Keywords:
Invariant means,
amenable groups,
mean ergodic theorem,
exposed points,
StoneČech compactification
Article copyright:
© Copyright 1971 American Mathematical Society
