The Bohr compactification of an abelian group as a quotient of its Stone-Čech compactification

2018/08/14 by Zlatoš, Pavol
#22C05 #43A60 #54H11 #FOS: Mathematics #General Topology (math.GN) #Primary 22A15 #Secondary 43A40

paper · doi:10.48550/arxiv.1808.04637

Abstract

We will prove that, for any abelian group G, the canonical (surjective and continuous) mapping \boldsymbolβG → \frak bG from the Stone-Čech compactification \boldsymbolβG of G to its Bohr compactfication \frak bG is a homomorphism with respect to the semigroup operation on \boldsymbolβG, extending the multiplication on G, and the group operation on \frak bG. Moreover, the Bohr compactification \frak bG is canonically isomorphic (both in algebraic and topological sense) to the quotient of \boldsymbolβG with respect to the least closed congruence relation on \boldsymbolβG merging all the Schur ultrafilters on G into the unit of G.

Related

• Bohr compactification and Chu duality of non-abelian locally compact groups
• The Bohr compactification of an arithmetic group
• Schur ultrafilters and Bohr compactifications of topological groups
• Bohr compactifications of algebras and structures
• Remarks on compactifications of pseudofinite groups