Functional analytic properties of topological semigroups and -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 be a topological semigroup, be the space of left uniformly continuous functions on , and be the set of multiplicative means on . If has a left invariant mean in the convex hull of , we associate with a *unique* finite group such that for any maximal proper closed left translation invariant ideal in , there exists a linear isometry mapping one-one onto the set of bounded real functions on . We also generalise some recent results of T. Mitchell and E. Granirer. In particular, we show that satisfies iff whenever is a jointly continuous action on a compact hausdorff space , there exists a nonempty finite subset of such that for all . Furthermore, a discrete semigroup satisfies iff whenever is an antirepresentation of as linear maps from a norm linear space into with for all , there exists a finite subset such that the distance (induced by the norm) of from linear span of in coincides with distance of from 0 for all .

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DOI:
https://doi.org/10.1090/S0002-9947-1970-0269772-9

Keywords:
Topological semigroups,
-extremely amenable,
amenable semigroup,
-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

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© Copyright 1970
American Mathematical Society