Maximally Almost Periodic Groups and a Theorem of Glicksberga

Abstract

ABSTRACT:It is well known that if G is a locally compact Abelian group (LCA group) with Bohr compactification (β(G), σ) then σ(G) is normal in β(G) and, by a beautiful theorem of Glicksberg, we have that A⊂σ(G) is compact if and only if σ–1(A) ⊂G is compact. The aim of this paper is to study maximally almost periodic (MAP) groups which have these properties and the results obtained are as follows. (1) If G is a σ‐compact locally compact MAP group with Bohr compactification (β(G), σ) and σ(G) is normal in β(G), then for each gεβ(G), the automorphism induced by σ and conjugation by g is actually a topological isomorphism. (2) A finite extension of a LCA group is a MAP group and it has the property that A⊂σ(G) is compact if and only if σ–1(A) ⊂G is compact, and (3) A discrete MAP group G with Bohr compactification (β(G), σ) satisfying both of the properties being considered must be Abelian by finite, i.e., a finite extension of an Abelian group.