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Locally Compact Groups, Invariant Means and the Centres of Compactifications
Author(s) -
Lau A. T.,
Milnes P.,
Pym J. S.
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610797005383
Subject(s) - locally compact space , mathematics , compactification (mathematics) , homomorphism , invariant (physics) , semigroup , locally compact group , bounded function , pure mathematics , algebraic number , discrete mathematics , compact space , topology (electrical circuits) , combinatorics , mathematical analysis , mathematical physics
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 ∞ ( 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, ∞), +), 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.