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The continuity of Arens' product on the Stone-Čech compactification of semigroups


Author: Nicholas Macri
Journal: Trans. Amer. Math. Soc. 191 (1974), 185-193
MSC: Primary 22A25
DOI: https://doi.org/10.1090/S0002-9947-1974-0382541-8
MathSciNet review: 0382541
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Abstract: A discrete semigroup is said to have the compact semigroup property (c.s.p.) [the compact semi-semigroup property (c.s.s.p.)] if the multiplication Arens' product, on its Stone-Čech compactification, is jointly [separately] $ {w^ \ast }$-continuous. We obtain an algebraic characterization of those semigroups which have c.s.p. by characterizing algebraically their almost periodic subsets. We show that a semigroup has c.s.p. if and only if each of its subsets is almost periodic. This characterization is employed to prove that for a cancellation semigroup to have c.s.p., it is necessary and sufficient that each of its countable subsets be almost periodic. We answer in the negative a heretofore open question--is c.s.p. equivalent to c.s.s.p.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0382541-8
Keywords: Arens' product, joint continuity, separate continuity, Stone-Čech compactification, almost periodic, weakly almost periodic
Article copyright: © Copyright 1974 American Mathematical Society

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