The exposed points of the set of invariant means on an ideal
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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 $\beta \mathbb {N}$.References
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Additional Information
- Tianxuan Miao
- Affiliation: Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada P7E 5E1
- Email: tmiao@thunder.lakeheadu.ca
- 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
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 3571-3579
- MSC (1991): Primary 43A07
- DOI: https://doi.org/10.1090/S0002-9939-98-04550-X
- MathSciNet review: 1468200