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Reductive compactifications of semitopological semigroups. (English) Zbl 1028.22005
Let $$S$$ be a semitopological semigroup. A pair $$(\psi ,X)$$ is called a semigroup compactification of $$S$$ if $$X$$ is a compact Hausdorff right topological semigroup, $$\psi :S\to X$$ is a continuous homomorphism with a dense image such that for every $$s\in S$$ the mapping $$x\mapsto\psi (s)x$$ is continuous. An enveloping semigroup of a semigroup compactification $$(\psi ,X)$$ determines a semigroup compactification of $$S$$ that is isomorphic to $$(\psi ,X)$$ if and only if $$X$$ is right reductive. Any semigroup compactification of $$S$$ is right reductive if $$sS$$ (or $$Ss$$) is dense in $$S$$ for some $$s\in S$$. A special semigroup compactification by continuous complex value functions from $$S$$ is defined.

##### MSC:
 22A20 Analysis on topological semigroups 43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
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