Functional analytic properties of topological semigroups and extreme amenability
Author:
Anthony Toming Lau
Journal:
Trans. Amer. Math. Soc. 152 (1970), 431439
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 oneone 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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 J. Sorenson, Existence of measures that are invariant under a semigroup of transformations, Thesis, Purdue University, Lafayette, Ind., 1966.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197002697729
PII:
S 00029947(1970)02697729
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
Article copyright:
© Copyright 1970
American Mathematical Society
