Bohr compactification and Chu duality of non-abelian locally compact groups
María V. Ferrer, S. Hernández
TL;DR
The paper investigates the relationship between the Bohr compactification $bG$ and Chu duality for locally compact groups, focusing on when $bG$ is 'small' in the non‑abelian setting. It develops the Chu bidual $G^{xx}$ and analyzes when $bG$ and $G^{xx}$ coincide, proving that this occurs iff every partial dual $G_n^x$ is discrete. The authors construct explicit non‑compact Chu reflexive groups with discrete partial duals (e.g., $G=H^{(I)}$ for a finite simple non‑abelian group $H$) and correct several statements in the literature, including the nuanced relationship between discreteness of $\widehat{G}_n$ and Chu duality. The results refine our understanding of duality and compactifications in the non‑abelian locally compact context and have implications for the structure of dual objects, vdW groups, and Kazhdan groups.
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.