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Transactions of the American Mathematical Society

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



Topological semigroups and representations. I

Author: James C. S. Wong
Journal: Trans. Amer. Math. Soc. 200 (1974), 89-109
MSC: Primary 22A20
MathSciNet review: 0369604
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Abstract: Let $ S$ be a topological semigroup (separately continuous multiplication) with identity and $ W(S)$ the Banach space of all weakly almost periodic functions on $ S$. It is well known that if $ S = G$ is a locally compact group, then $ W(G)$ always has a (unique) invariant mean. In other words, there exists $ m \in W{(G)^ \ast }$ such that $ \vert\vert m\vert\vert = m(1) = 1$ and $ m({l_s}f) = m({r_s}f) = m(f)$ for any $ s \in G,f \in W(G)$ where $ {l_s}f(t) = f(st)$ and $ {r_s}f(t) = f(ts),t \in S$ The main purpose of this paper is to present several characterisations (functional analytic and algebraic) of the existence of a left (right) invariant mean on $ W(S)$ In particular, we prove that $ W(S)$ has a left (right) invariant mean iff a certain compact topological semigroup $ p{(S)^\omega }$ (to be defined) associated with $ S$ contains a right (left) zero. Other results in this direction are also obtained.

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Keywords: Topological semigroups, compact semigroups, convex semigroups, almost periodic functions, almost periodic compactifications, invariant means, representations of topological semigroups, fixed point properties
Article copyright: © Copyright 1974 American Mathematical Society

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