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Compactifications of locally compact groups and quotients*

Published online by Cambridge University Press:  24 October 2008

A. T. Lau ,
P. Milnes  and
J. S. Pym
A. T. Lau
Affiliation:
Edmonton, Alberta T6G 2G1; London, Ontario N6A 5B7, Canada; and Sheffield S10 2TN
P. Milnes
Affiliation:
Edmonton, Alberta T6G 2G1; London, Ontario N6A 5B7, Canada; and Sheffield S10 2TN
J. S. Pym
Affiliation:
Edmonton, Alberta T6G 2G1; London, Ontario N6A 5B7, Canada; and Sheffield S10 2TN
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Abstract

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 (ψ, X) of G by Nψ = ψ(N) ⊂ X; this is achieved in a number of cases for which we can establish that μNψ ⊂ Nψ μ for all μ ∈ 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 GN and can be identified algebraically with Nx GN 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 GN. When GN 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 (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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 3 , November 1994 , pp. 451 - 463
Copyright
Copyright © Cambridge Philosophical Society 1994

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