The exposed points of the set of invariant means on an ideal
Author:
Tianxuan Miao
Journal:
Proc. Amer. Math. Soc. 126 (1998), 35713579
MSC (1991):
Primary 43A07
MathSciNet review:
1468200
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Abstract: Let be a compact locally compact nondiscrete group and let be a invariant ideal of . We denote the set of left invariant means on that are zero on (i.e. for all ) by . We show that, when is amenable as a discrete group and the closed invariant subset of the spectrum of corresponding to is a set, is very large in the sense that every nonempty subset of contains a norm discrete copy of , where is the Stone compactification of the set of positive integers with the discrete topology. In particular, we prove that has no exposed points in this case and every nonempty subset of the set of left invariant means on contains a norm discrete copy of .
 1.
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
(51 #1343)
 2.
E.
Granirer, On finite equivalent invariant measures for semigroups of
transformations, Duke Math. J. 38 (1971),
395–408. MR 0283171
(44 #404)
 3.
Edmond
Granirer, Criteria for compactness and for
discreteness of locally compact amenable groups, Proc. Amer. Math. Soc. 40 (1973), 615–624. MR 0340962
(49 #5712), http://dx.doi.org/10.1090/S00029939197303409628
 4.
Frederick
P. Greenleaf, Invariant means on topological groups and their
applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand
Reinhold Co., New YorkToronto, Ont.London, 1969. MR 0251549
(40 #4776)
 5.
Tianxuan
Miao, The exposed points of the set of
invariant means, Trans. Amer. Math. Soc.
347 (1995), no. 4,
1401–1408. MR 1260174
(95g:43003), http://dx.doi.org/10.1090/S00029947199512601742
 6.
Tianxuan
Miao, On the sizes of the sets of invariant means, Illinois J.
Math. 36 (1992), no. 1, 53–72. MR 1133770
(93a:43001)
 7.
Alan
L. T. Paterson, Amenability, Mathematical Surveys and
Monographs, vol. 29, American Mathematical Society, Providence, RI,
1988. MR
961261 (90e:43001)
 8.
Alan
L. T. Paterson, Amenability, Mathematical Surveys and
Monographs, vol. 29, American Mathematical Society, Providence, RI,
1988. MR
961261 (90e:43001)
 9.
Michel
Talagrand, Géométrie des simplexes de moyennes
invariantes, J. Funct. Anal. 34 (1979), no. 2,
304–337 (French). MR 552708
(80k:43002), http://dx.doi.org/10.1016/00221236(79)900375
 10.
Michel
Talagrand, Invariant means on an ideal,
Trans. Amer. Math. Soc. 288 (1985),
no. 1, 257–272. MR 773060
(86e:43005), http://dx.doi.org/10.1090/S00029947198507730602
 11.
Michel
Talagrand, Moyennes invariantes s’annulant sur des
idéaux, Compositio Math. 42 (1980/81),
no. 2, 213–216 (French). MR 596876
(82b:43003)
 1.
 E. E. Granirer, Exposed points of convex sets and weak sequential convergence, Mem. Amer. Math. Soc. 123 (1972). MR 51:1343
 2.
 , On finite equivalent invariant measures for semigroups of transformations, Duke Math. J., 38 (1971), 395408. MR 44:404
 3.
 , Criteria for compactness and for discreteness of locally compact amenable groups, Proc. Amer. Math. Soc. 40 (1973), 615624. MR 49:5712
 4.
 F. P. Greenleaf, Invariant Means on Topological Groups, Van Nostrand, New York, 1969. MR 40:4776
 5.
 T. Miao, The exposed points of the set of invariant means, Trans. Amer. Math. Soc. 347 (1995), 14011408. MR 95g:43003
 6.
 , On the sizes of the sets of invariant means, Illinois J. Math. 36 (1992), 5372. MR 93a:43001
 7.
 A. L. T. Paterson, Amenability, Amer. Math. Soc., Providence, Rhode Island, 1988. MR 90e:43001
 8.
 J. P. Pier, Amenable Locally Compact Groups, Wiley, New York, 1984. MR 90e:43001
 9.
 M. Talagrand, Géométrie des simplexes de moyennes invariantes, J. Funct. Anal. 34 (1979), 304337. MR 80k:43002
 10.
 , Invariant means on an ideal, Trans. Amer. Math. Soc. 288 (1985), 257272. MR 86e:43005
 11.
 , Moyennes invariantes s'annulant sur des ideaux, Compositio Math. 42 (1981), 213216. MR 82b:43003
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Additional Information
Tianxuan Miao
Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada P7E 5E1
Email:
tmiao@thunder.lakeheadu.ca
DOI:
http://dx.doi.org/10.1090/S000299399804550X
PII:
S 00029939(98)04550X
Keywords:
Locally compact groups,
amenable groups,
invariant means,
invariant ideals,
exposed points
Received by editor(s):
December 12, 1996
Received by editor(s) in revised form:
April 20, 1997
Additional Notes:
This research is supported by an NSERC grant.
Communicated by:
J. Marshall Ash
Article copyright:
© Copyright 1998
American Mathematical Society
