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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
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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 $\beta \mathbb{N}$.

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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

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