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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Functional analytic properties of topological semigroups and $ n$-extreme amenability

Author: Anthony To-ming Lau
Journal: Trans. Amer. Math. Soc. 152 (1970), 431-439
MSC: Primary 22.05; Secondary 46.00
MathSciNet review: 0269772
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Abstract: Let $ S$ be a topological semigroup, $ \operatorname{LUC} (S)$ be the space of left uniformly continuous functions on $ S$, and $ \Delta (S)$ be the set of multiplicative means on $ \operatorname{LUC} (S)$. If $ ( \ast )\operatorname{LUC} (S)$ has a left invariant mean in the convex hull of $ \Delta (S)$, we associate with $ S$ a unique finite group $ G$ such that for any maximal proper closed left translation invariant ideal $ I$ in $ \operatorname{LUC} (S)$, there exists a linear isometry mapping $ \operatorname{LUC} (G)/I$ one-one onto the set of bounded real functions on $ G$. We also generalise some recent results of T. Mitchell and E. Granirer. In particular, we show that $ S$ satisfies $ ( \ast )$ iff whenever $ S$ is a jointly continuous action on a compact hausdorff space $ X$, there exists a nonempty finite subset $ F$ of $ X$ such that $ sF = F$ for all $ s \in S$. Furthermore, a discrete semigroup $ S$ satisfies $ ( \ast )$ iff whenever $ \{ {T_s};s \in S\} $ is an antirepresentation of $ S$ as linear maps from a norm linear space $ X$ into $ X$ with $ \vert\vert{T_s}\vert\vert \leqq 1$ for all $ s \in S$, there exists a finite subset $ \sigma \subseteq S$ such that the distance (induced by the norm) of $ x$ from $ {K_X} =$   linear span of $ \{ x - {T_s}x;x \in X,s \in S\} $ in $ X$ coincides with distance of $ O(\sigma ,x) = \{ (1/\vert\sigma \vert)\sum\nolimits_{a \in \sigma } {{T_{at}}(x);t \in S\} } $ from 0 for all $ x \in X$.

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Keywords: Topological semigroups, $ n$-extremely amenable, amenable semigroup, $ n$-extremely amenable semigroup, uniformly continuous functions, jointly continuous actions, multiplicative means, invariant means, maximal translation invariant closed ideal, finite intersection property, right ideal, group homomorphisms, locally compact groups, fixed points, point measure
Article copyright: © Copyright 1970 American Mathematical Society