## Invariant means on an ideal

HTML articles powered by AMS MathViewer

- by Michel Talagrand PDF
- Trans. Amer. Math. Soc.
**288**(1985), 257-272 Request permission

## Abstract:

Let $G$ be a compact abelian group and $Q$ an invariant ideal of ${L^\infty }(G)$. Let ${M_Q}$ be the set of invariant means $\nu$ on ${L^\infty }(G)$ that are zero on $Q$, that is $\nu ({\chi _A}) = 1$ for ${\chi _A} \in Q$. We show that ${M_Q}$ is very large in the sense that a nonempty ${G_\delta }$ subset of ${M_Q}$ must contain a copy of $\beta {\mathbf {N}}$. Let ${E_Q}$ be the set of extreme points of ${M_Q}$. We show that its closure is very small in the sense that it contains no nonempty ${G_\delta }$ of ${M_Q}$. We also show that ${E_Q}$ is topologically very irregular in the sense that it contains no nonempty ${G_\delta }$ of its closure. The proofs are based on delicate constructions which rely on combinatorial type properties of abelian groups. Assume now that $G$ is locally compact, noncompact, nondiscrete and countable at infinity. Let $M$ be the set of invariant means on ${L^\infty }(G)$ and ${M_t}$, the set of topologically invariant means. We show that ${M_t}$ is very small in $M$. More precisely, each nonempty ${G_\delta }$ subset of $M$ contains a $\nu$ such that $\nu (f) = 1$ for some $f \in C(G)$]> with $0 \leqslant f \leqslant 1$ and the support of $f$ has a finite measure. Under continuum hypothesis, we also show that there exists points in ${M_t}$ which are extremal in $M$ (but, in general, ${M_t}$ is not a face of $M$, that is, not all the extreme points of ${M_t}$ are extremal in $M$).## References

- Erik M. Alfsen,
*Compact convex sets and boundary integrals*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57, Springer-Verlag, New York-Heidelberg, 1971. MR**0445271**
N. Bourbaki, - George Converse, Isaac Namioka, and R. R. Phelps,
*Extreme invariant positive operators*, Trans. Amer. Math. Soc.**137**(1969), 375–385. MR**243370**, DOI 10.1090/S0002-9947-1969-0243370-7 - W. R. Emerson and F. P. Greenleaf,
*Covering properties and Følner conditions for locally compact groups*, Math. Z.**102**(1967), 370–384. MR**220860**, DOI 10.1007/BF01111075 - Edmond E. Granirer,
*Exposed points of convex sets and weak sequential convergence*, Memoirs of the American Mathematical Society, No. 123, American Mathematical Society, Providence, R.I., 1972. Applications to invariant means, to existence of invariant measures for a semigroup of Markov operators etc. . MR**0365090** - Frederick P. Greenleaf,
*Invariant means on topological groups and their applications*, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR**0251549**
P. A. Meyer, - Joseph Max Rosenblatt,
*Invariant means and invariant ideals in $L_{\infty }(G)$ for a locally compact group $G$*, J. Functional Analysis**21**(1976), no. 1, 31–51. MR**0397304**, DOI 10.1016/0022-1236(76)90027-6 - Michel Talagrand,
*Géométrie des simplexes de moyennes invariantes*, J. Functional Analysis**34**(1979), no. 2, 304–337 (French). MR**552708**, DOI 10.1016/0022-1236(79)90037-5
—,

*Topologie générale*, Chapitre 7, Hermann, Paris, 1947. G. Choquet,

*Lectures on analysis*, Benjamin, New York, 1969.

*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.

*Moyennes invariantes s’annulant sur des ideaux*, Compositio Math.

**42**(1981), 213-216.

## Additional Information

- © Copyright 1985 American Mathematical Society
- 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