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Maximally Almost Periodic Groups and a Theorem of Glicksberg a
Author(s) -
WU TA SUN,
RIGGINS LEMUEL
Publication year - 1996
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1996.tb49188.x
Subject(s) - locally compact space , mathematics , abelian group , compactification (mathematics) , bohr model , automorphism , topological group , locally compact group , g module , pure mathematics , elementary abelian group , discrete mathematics , combinatorics , topology (electrical circuits) , physics , quantum mechanics
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 g εβ( G ), the automorphism induced by σ and conjugation by g is actually a topological isomorphism. (2) A finite extension of a L CA 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.