Compactifications of locally compact groups and quotients

Abstract

AbstractLetNbe a compact normal subgroup of a locally compact groupG. One of our goals here is to determine when and how a given compactificationYofG/Ncan be realized as a quotient of the analogous compactification (ψ,X) ofGbyNψ= ψ(N) ⊂X; this is achieved in a number of cases for which we can establish that μNψ⊂Nψμ for all μ ∈XA question arises naturally, ‘Can the latter containment be proper?’ With an example, we give a positive answer to this question.The groupGis an extension ofNbyGNand can be identified algebraically withNx GNwhen this product is given the Schreier multiplication, and for our further results we assume that we can also identify G topologically withN x GN. WhenGNis discrete andXis the compactification ofGcoming from the left uniformly continuous functions, we are able to show thatXis an extension ofNby(GN)(X≅N x (G/N))even when G is not a semidirect product. Examples are given to illustrate the theory, and also to show its limitations.