Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The exposed points of the set
of invariant means on an ideal

Author: Tianxuan Miao
Journal: Proc. Amer. Math. Soc. 126 (1998), 3571-3579
MSC (1991): Primary 43A07
MathSciNet review: 1468200
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a $\sigma $-compact locally compact nondiscrete group and let $Q$ be a $G$-invariant ideal of $L^{\infty }(G)$. We denote the set of left invariant means $ m$ on $L^{\infty }(G)$ that are zero on $Q$ (i.e. $m(f) = 0$ for all $f\in Q$) by $LIM_{Q}$. We show that, when $G$ is amenable as a discrete group and the closed $G$-invariant subset of the spectrum of $L^{\infty }(G)$ corresponding to $Q$ is a $G_{\delta }$-set, $LIM_{Q}$ is very large in the sense that every nonempty $G_{\delta }$-subset of $LIM_{Q}$ contains a norm discrete copy of $\beta \mathbb{N}$, where $\beta \mathbb{N}$ is the Stone-$\mathrm{\check{C}ech}$ compactification of the set $\mathbb{N} $ of positive integers with the discrete topology. In particular, we prove that $LIM_{Q}$ has no exposed points in this case and every nonempty $G_{\delta }$-subset of the set of left invariant means on $L^{\infty }(G)$ contains a norm discrete copy of $\beta \mathbb{N}$.

References [Enhancements On Off] (What's this?)

  • 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), 395-408. MR 44:404
  • 3. -, Criteria for compactness and for discreteness of locally compact amenable groups, Proc. Amer. Math. Soc. 40 (1973), 615-624. 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), 1401-1408. MR 95g:43003
  • 6. -, On the sizes of the sets of invariant means, Illinois J. Math. 36 (1992), 53-72. 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), 304-337. MR 80k:43002
  • 10. -, Invariant means on an ideal, Trans. Amer. Math. Soc. 288 (1985), 257-272. MR 86e:43005
  • 11. -, Moyennes invariantes s'annulant sur des ideaux, Compositio Math. 42 (1981), 213-216. MR 82b:43003

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 43A07

Retrieve articles in all journals with MSC (1991): 43A07

Additional Information

Tianxuan Miao
Affiliation: Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada P7E 5E1

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

American Mathematical Society