Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
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 Free Access

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?)

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

PII: S 0002-9939(98)04550-X
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

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia