LOCALLY COMPACT-GROUPS, INVARIANT-MEANS AND THE CENTERS OF COMPACTIFICATIONS

This paper brings together two apparently unrelated results about loca lly compact groups G by giving them a common proof. The first concerns the number of topologically left invariant means on L-infinity(G), wh ile the second slates that the topological centre of the largest semig roup compactification of G is simply G itself. On the way, we introduc e as vital tools some new compactifications of the half line ([0, infi nity), +), we produce a right invariant pseudometric on a compactly ge nerated G for which the bounded sets are precisely the relatively comp act sets, and we receive striking confirmation that some algebraic pro perties of semigroups can be transferred by maps which are quite far f rom being homomorphisms.