The continuity of Arens’ product on the Stone-Čech compactification of semigroups
NICHOLAS A. MACRI · Transactions of the American Mathematical Society · 1974
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 ∗ {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.