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Journal of the London Mathematical Society 1997 56(1):77-90; doi:10.1112/S0024610797005383
© 1997 by London Mathematical Society
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© The London Mathematical Society

Locally Compact Groups, Invariant Means and the Centres of Compactifications

A. T. Lau, P. Milnes and J. S. Pym

Department of Mathematics, University of Alberta Edmonton, Alberta T6G 2G1, Canada
Department of Mathematics, University of Western Ontario London, Ontario N6A 3K7, Canada
Department of Pure Mathematics, University of Sheffield Hicks Building, Sheffield S3 7RH

Received 5 December 1994. Revision received 20 July 1995.

This paper brings together two apparently unrelated results about locally compact groups G by giving them a common proof. The first concerns the number of topologically left invariant means on L{infty}(G), while the second states that the topological centre of the largest semigroup compactification of G is simply G itself. On the way, we introduce as vital tools some new compactifications of the half line ([0, {infty}), +), we produce a right invariant pseudometric on a compactly generated G for which the bounded sets are precisely the relatively compact sets, and we receive striking confirmation that some algebraic properties of semigroups can be transferred by maps which are quite far from being homomorphisms.


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