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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Invariant means on locally compact semigroups


Author: James C. S. Wong
Journal: Proc. Amer. Math. Soc. 31 (1972), 39-45
DOI: https://doi.org/10.1090/S0002-9939-1972-0289708-1
MathSciNet review: 0289708
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Abstract: Let $ G$ be a locally compact semigroup (jointly continuous semigroup operation), $ M(G)$ the algebra of all bounded regular Borel measures on $ G$ (with convolution as multiplication), $ E$ a separated locally convex space and $ S$ a compact convex subset of $ E$. We show that there is a left invariant mean on the space $ {\text{LUC}}(G)$ of all bounded left uniformly continuous functions on $ G$ iff $ G$ has the following fixed point property: For any bilinear mapping $ T:M(G) \times E \to E$ (denoted by $ (\mu ,s) \to {T_\mu }(s)$) such that (a) $ {T_\mu }(S) \subset S$ for any $ \mu \geqq 0,\vert\vert\mu \vert\vert = 1$, (b) $ {T_{\mu \ast\nu }} = {T_\mu } \circ {T_\nu }$ for any $ \mu ,\nu \in M(G)$, (c) $ {T_\mu }:S \to S$ is continuous for any $ \mu \geqq 0,\vert\vert\mu \vert\vert = 1$, and $ {\text{(d)}}\mu \to {T_\mu }(s)$ is continuous for each $ s \in S$ when $ M(G)$ has the topology induced by the seminorms $ {p_f}(\mu ) = \vert\int {fd\mu \vert} ,f \in {\text{LUC}}(G)$, there is some $ {s_0} \in S$ such that $ {T_\mu }({s_0}) = {s_0}$ for any $ \mu \geqq 0,\vert\vert\mu \vert\vert = 1$.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0289708-1
Keywords: Locally compact semigroups, fixed point properties, invariant means, convolution algebras, left uniformly continuous functions
Article copyright: © Copyright 1972 American Mathematical Society