Extension of additive functionals and semicharacters on commutative semigroups

Let S be a commutative semigroup and S_o a subsemigroup. The present paper establishes a necessary and sufficient condition in order that any real additive functional ϕ₀ defined on S_o and dominated there by a real subadditive functional p defined on S, admit an additive extension to S such that ϕ≤p. This result is a strengthening of a result of R.Kaufman 3]. From this easily follow recent results of Kobayashi 5] and Putcha, Tamura 9] on extension of semigroup homomorphisms. Also a result of Ross on extension of semicharacters can be deduced from this result. As an application, the existence of continuous semicharacters on an open subsemigroup of an abelian topological group is derived. This too, generalizes previous results (Gleason in 10]). Supported in part by the Danish Natural Science Research Council