Remarks on the quantum Bohr compactification

Abstract

The category of locally compact quantum groups can be described as either Hopf $*$-homomorphisms between universal quantum groups, or as bicharacters on reduced quantum groups. We show how Sołtan’s quantum Bohr compactification can be used to construct a “compactification” in this category. Depending on the viewpoint, different C$^{*}$-algebraic compact quantum groups are produced, but the underlying Hopf $*$-algebras are always, canonically, the same. We show that a complicated range of behaviours, with C$^{*}$-completions between the reduced and universal level, can occur even in the cocommutative case, thus answering a question of Sołtan. We also study such compactifications from the perspective of (almost) periodic functions. We give a definition of a periodic element in $L^{\infty}(\mathbb{G})$, involving the antipode, which allows one to compute the Hopf $*$-algebra of the compactification of $\mathbb{G} $; we later study when the antipode assumption can be dropped. In the cocommutative case, we make a detailed study of Runde’s notion of a completely almost periodic functional—with a slight strengthening, we show that for [SIN] groups this does recover the Bohr compactification of $\hat{G}$.