Recapturing semigroup compactifications of a group from those of its closed normal subgroups

Abstract

We know that if S is a subsemigroup of a semitopological semigroup T, and 𝔉 stands for one of the spaces 𝒜𝒫,𝒲𝒜𝒫,𝒮𝒜𝒫,𝒟 or ℒ𝒞, and (ϵ,T𝔉) denotes the canonical 𝔉-compactification of T, where T has the property that 𝔉(S)=𝔉(T)|s, then (ϵ|s,ϵ(S)¯) is an 𝔉-compactification of S. In this paper, we try to show the converse of this problem when T is a locally compact group and S is a closed normal subgroup of T. In this way we construct various semigroup compactifications of T from the same type compactifications of S.