Bohr compactification and Chu duality of non-abelian locally compact groups

TL;DR

This paper investigates the properties of the Bohr compactification of non-abelian locally compact groups, characterizing when it is small and relating it to Chu duality, correcting previous misconceptions in the literature.

Contribution

It provides a characterization of when the Bohr compactification of a locally compact group is isomorphic to its Chu or unitary quasi-dual, clarifying the structure for non-abelian groups.

Findings

Characterization of when the Bohr compactification is small

Conditions under which it is isomorphic to the Chu dual

Correction of previous inaccuracies in the literature

Abstract

The \emph{Bohr compactification} of an arbitrary topological group $G$ is defined as the group compactification $(bG,b)$ with the following universal property: for every continuous homomorphism $h$ from $G$ into a compact group $K$ there is a continuous homomorphism $h^{b}$ from $bG$ into $K$ extending $h$ in the sense that $h=h^b \circ b$. The Bohr compactification $(bG,b)$ is the unique (up to equivalence) largest compactification of $G$. Although, for locally compact Abelian groups, the Bohr compactification is a big monster, for non-Abelian groups the situation is much more interesting and it can be said that all options are possible. Here we are interested in locally compact groups whose Bohr compactification is \emph{small}. Among other results, we characterize when the Bohr the Bohr compactification of a locally compact group is topologically isomorphic to its Chu or unitary quasi-dual. Our results fixe some incorrect statements appeared in the literature.