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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On a geometric property of the set of invariant means on a group

Author: Ching Chou
Journal: Proc. Amer. Math. Soc. 30 (1971), 296-302
MSC: Primary 46.80; Secondary 42.00
MathSciNet review: 0283584
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Abstract: If G is a discrete group and $ x \in G$ then $ x^\sim$ denotes the homeomorphism of $ \beta G$ onto $ \beta G$ induced by left multiplication by x. A subset K of $ \beta G$ is said to be invariant if it is closed, nonempty and $ x^\sim \emptyset K \subset K$ for each $ x \in G$. Let $ ML(G)$ denote the set of left invariant means on G. (They can be considered as measures on $ \beta G$.)

Let G be a countably infinite amenable group and let K be an invariant subset of $ \beta G$. Then the nonempty $ {w^ \ast }$-compact convex set $ M(G,K) = \{ \phi \in ML(G):{\text{suppt}}\phi \subset K\} $ has no exposed points (with respect to $ {w^ \ast }$-topology). Therefore, it is infinite dimensional.

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PII: S 0002-9939(1971)0283584-8
Keywords: Invariant means, amenable groups, mean ergodic theorem, exposed points, Stone-Čech compactification
Article copyright: © Copyright 1971 American Mathematical Society