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

Macri, Nicholas

doi:10.1090/S0002-9947-1974-0382541-8

The continuity of Arens’ product on the Stone-Čech compactification of semigroups
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Trans. Amer. Math. Soc. 191 (1974), 185-193 Request permission

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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