COMPACTIFICATIONS OF LOCALLY COMPACT-GROUPS AND QUOTIENTS

Let N be a compact normal subgroup of a locally compact group G. One of our goals here is to determine when and how a given compactification Y of G/N can be realized as a quotient of the analogous compactification (psi, X) of G by N(psi) = psi(N) subset-of X; this is achieved in a number of cases for which we can establish that muN(psi) subset-of N(psi)mu for all mu is-an-element-of X. A question arises naturally, 'Can the latter containment be proper?' With an example, we give a positive answer to this question. The group G is an extension of N by G/N and can be identified algebraically with N x G/N when this product is given the Schreier multiplication, and for our further results we assume that we can also identify G topologically with N x G/N. When G/N is discrete and X is the compactification of G coming from the left uniformly continuous functions, we are able to show that X is an extension of N by beta(G/N) (X congruent-to N x beta (G/N)) even when G is not a semidirect product. Examples are given to illustrate the theory, and also to show its limitations.