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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Invariant means on an ideal


Author: Michel Talagrand
Journal: Trans. Amer. Math. Soc. 288 (1985), 257-272
MSC: Primary 43A07; Secondary 46A55
MathSciNet review: 773060
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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 <![CDATA[ $ 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$).


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0773060-2
PII: S 0002-9947(1985)0773060-2
Keywords: Invariant mean, invariant ideal, extreme point, exposed point, geometry of the set of invariant means
Article copyright: © Copyright 1985 American Mathematical Society