© 1999 by London Mathematical Society
© The London Mathematical Society
On the Structure of Minimal Left Ideals in the Largest Compactification of a Locally Compact Group
Department of Mathematics, University of Alberta Edmonton, Alberta T6G 2G1, Canada
Department of Mathematics, University of Western Ontario London, Ontario N6A 5B7, Canada
Department of Pure Mathematics, University of Sheffield Hicks Building, Sheffield S3 7RH
Received 25 November 1996. Revision received 12 February 1997.
This paper is centred around a single question: can a minimal left ideal L in GLUC, the largest semi-group compactification of a locally compact group G, be itself algebraically a group? Our answer is no (unless G is compact). In deriving this conclusion, we obtain for nearly all groups the stronger result that no maximal subgroup in L can be closed. A feature of our work is that completely different techniques are required for the connected and totally disconnected cases. For the former, we can rely on the extensive structure theory of connected, non-compact, locally compact groups to derive the solution from the commutative case, using some reduction lemmas. The latter directly involves topological dynamics; we construct a compact space and an action of G on it which has pathological properties. We obtain other results as tools towards our main goal or as consequences of our methods. Thus we find an extension to earlier work on the relationship between minimal left ideals in GLUC and HLUC when H is a closed subgroup of G with G/H compact. We show that the distal compactification of G is finite if and only if the almost periodic compactification of G is finite. Finally, we use our methods to show that there is no finite subset of GLUC invariant under the right action of G when G is an almost connected group or an IN-group.