The exposed points of the set of invariant means
Author:
Tianxuan Miao
Journal:
Trans. Amer. Math. Soc. 347 (1995), 14011408
MSC:
Primary 43A07
MathSciNet review:
1260174
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Abstract: Let be a compact infinite locally compact group, and let be the set of left invariant means on . We prove in this paper that if is amenable as a discrete group, then has no exposed points. We also give another proof of the Granirer theorem that the set of invariant means on has no exposed points, where is an amenable countable group acting ergodically as measurepreserving transformations on a nonatomic probability space .
 [1]
C. Chou, On a geometric property of the set of invariant means on a group, Proc. Amer. Math. Soc. 30 (1971), 296302.
 [2]
, Ergodic group actions with nonunique invariant means, Proc. Amer. Math. Soc. 100 (1987), 647650.
 [3]
A. del Junco and J. Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann. 245 (1979), 185197.
 [4]
E.E. Granirer, Exposed points of convex sets and weak sequential convergence, Mem. Amer. Math. Soc., vol. 123, Amer. Math. Soc., Providence, RI, 1972.
 [5]
, Geometric and topological properties of certain compact convex sets which arise from the study of invariant means, Canad. J. Math. 37 (1985), 107121.
 [6]
, Geometric and topological properties of certain compact convex subsets of double duals of Banach spaces, which arise from the study of invariant means, Illinois J. Math. 30 (1986), 148174.
 [7]
, Criteria for compactness and for discreteness of locally compact amenable groups, Proc. Amer. Math. Soc. 40 (1973), 615624.
 [8]
, On finite equivalent invariant measures for semigroups of transformations, Duke Math. J. 38 (1971), 395408.
 [9]
F.P. Greenleaf, Invariant means on topological groups, Van Nostrand, New York, 1969.
 [10]
A.L.T. Paterson, Amenability, Amer. Math. Soc., Providence, RI, 1988.
 [11]
J.P. Pier, Amenable locally compact groups, Wiley, New York, 1984.
 [12]
J.M. Rosenblatt, Invariant means and invariant ideals in for a locally compact group , J. Funct. Anal. 21 (1976), 3151.
 [13]
, Uniqueness of invariant means for measure preserving transformations, Trans. Amer. Math. Soc. 265 (1981), 623636.
 [14]
M. Talagrand, Géométrie des simplexes de moyennes invariantes, J. Funct. Anal. 34 (1979), 304337.
 [15]
Z. Yang, Exposed points of left invariant means, Pacific J. Math. 125 (1986), 487494.
 [1]
 C. Chou, On a geometric property of the set of invariant means on a group, Proc. Amer. Math. Soc. 30 (1971), 296302.
 [2]
 , Ergodic group actions with nonunique invariant means, Proc. Amer. Math. Soc. 100 (1987), 647650.
 [3]
 A. del Junco and J. Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann. 245 (1979), 185197.
 [4]
 E.E. Granirer, Exposed points of convex sets and weak sequential convergence, Mem. Amer. Math. Soc., vol. 123, Amer. Math. Soc., Providence, RI, 1972.
 [5]
 , Geometric and topological properties of certain compact convex sets which arise from the study of invariant means, Canad. J. Math. 37 (1985), 107121.
 [6]
 , Geometric and topological properties of certain compact convex subsets of double duals of Banach spaces, which arise from the study of invariant means, Illinois J. Math. 30 (1986), 148174.
 [7]
 , Criteria for compactness and for discreteness of locally compact amenable groups, Proc. Amer. Math. Soc. 40 (1973), 615624.
 [8]
 , On finite equivalent invariant measures for semigroups of transformations, Duke Math. J. 38 (1971), 395408.
 [9]
 F.P. Greenleaf, Invariant means on topological groups, Van Nostrand, New York, 1969.
 [10]
 A.L.T. Paterson, Amenability, Amer. Math. Soc., Providence, RI, 1988.
 [11]
 J.P. Pier, Amenable locally compact groups, Wiley, New York, 1984.
 [12]
 J.M. Rosenblatt, Invariant means and invariant ideals in for a locally compact group , J. Funct. Anal. 21 (1976), 3151.
 [13]
 , Uniqueness of invariant means for measure preserving transformations, Trans. Amer. Math. Soc. 265 (1981), 623636.
 [14]
 M. Talagrand, Géométrie des simplexes de moyennes invariantes, J. Funct. Anal. 34 (1979), 304337.
 [15]
 Z. Yang, Exposed points of left invariant means, Pacific J. Math. 125 (1986), 487494.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199512601742
PII:
S 00029947(1995)12601742
Keywords:
Locally compact groups,
amenable groups,
invariant means,
the exposed points
Article copyright:
© Copyright 1995
American Mathematical Society
