Invariant means on an ideal

Author:
Michel Talagrand

Journal:
Trans. Amer. Math. Soc. **288** (1985), 257-272

MSC:
Primary 43A07; Secondary 46A55

DOI:
https://doi.org/10.1090/S0002-9947-1985-0773060-2

MathSciNet review:
773060

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a compact abelian group and an invariant ideal of . Let be the set of invariant means on that are zero on , that is for . We show that is very large in the sense that a nonempty subset of must contain a copy of . Let be the set of extreme points of . We show that its closure is very small in the sense that it contains no nonempty of . We also show that is topologically very irregular in the sense that it contains no nonempty of its closure. The proofs are based on delicate constructions which rely on combinatorial type properties of abelian groups.

Assume now that is locally compact, noncompact, nondiscrete and countable at infinity. Let be the set of invariant means on and , the set of topologically invariant means. We show that is very small in . More precisely, each nonempty subset of contains a such that for some ]> with <![CDATA[ and the support of has a finite measure. Under continuum hypothesis, we also show that there exists points in which are extremal in (but, in general, is not a face of , that is, not all the extreme points of are extremal in ).

**[1]**E. M. Alfsen,*Compact convex sets and boundary integrals*, Springer-Verlag, Berlin, 1971. MR**0445271 (56:3615)****[2]**N. Bourbaki,*Topologie générale*, Chapitre 7, Hermann, Paris, 1947.**[3]**G. Choquet,*Lectures on analysis*, Benjamin, New York, 1969.**[4]**G. Converse, I. Namioka and R. R. Phelps,*Extreme invariant positive operators*, Trans. Amer. Math. Soc.**131**(1966), 376-385. MR**0243370 (39:4692)****[5]**W. R. Emerson and F. P. Greenleaf,*Covering properties and Folner conditions for locally compact groups*, Math. Z.**102**(1967), 370-384. MR**0220860 (36:3912)****[6]**E. Granirer,*Exposed points of convex sets and weak sequential convergence*, Mem. Amer. Math. Soc. No. 123 (1972). MR**0365090 (51:1343)****[7]**F. P. Greenleaf,*Invariant means on topological groups*, Van Nostrand Math. Stud., No. 16, Van Nostrand, Princeton, N. J., 1969. MR**0251549 (40:4776)****[8]**P. A. Meyer,*Limites médiales, d'après Mokobodzki*, Séminaire de Probabilités VII (Université de Strasbourg, 1973), Lecture Notes in Math., vol. 321, Springer-Verlag, Berlin and New York.**[9]**J. M. Rosenblatt,*Invariant means and invariant ideals in**for a locally compact group*, J. Funct. Anal.**21**(1976), 31-51. MR**0397304 (53:1163)****[10]**M. Talagrand.*Géométrie des Simplexes de moyennes invariantes*, J. Funct. Anal.**34**(1979), 304-337. MR**552708 (80k:43002)****[11]**-,*Moyennes invariantes s'annulant sur des ideaux*, Compositio Math.**42**(1981), 213-216.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
43A07,
46A55

Retrieve articles in all journals with MSC: 43A07, 46A55

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0773060-2

Keywords:
Invariant mean,
invariant ideal,
extreme point,
exposed point,
geometry of the set of invariant means

Article copyright:
© Copyright 1985
American Mathematical Society