Some Facets of an Lca Group inside Its Bohr Compactification

Let be a non-discrete LCA (locally compact abelian) group, its dual. ( ) ∧, where denotes discrete, is the Bohr compactification of , denoted by . There exists a continuous group isomorphism ί: → of onto a proper dense subgroup of . This subgroup to be denoted by ί is the main object of our study. A subgroup of is pure if ∩ = for each positive integer . We define to be strongly pure if is pure and [ ] = [ ] for each positive integer , where [ ] = { ∈ : = 0}. is (weakly) essential if every nonzero (closed) subgroup of has nonzero intersection with . A proper subgroup is totally dense in if ∩ is dense in for each closed subgroup of . We find conditions on or or both so that ί as a subgroup of may be: pure or strongly pure; essential or weakly essential; totally dense; maximal torsion; minimal dense;