Remarks on the Quantum Bohr Compactification

TL;DR

This paper explores the quantum Bohr compactification within the framework of locally compact quantum groups, revealing new behaviors and connections to almost periodic functions, and clarifying the structure of various C*-algebraic compact quantum groups.

Contribution

It demonstrates how Sołtan's quantum Bohr compactification constructs a canonical compactification in the quantum group category, clarifies behaviors of C*-completions, and links to almost periodic functions.

Findings

Different C*-algebraic compact quantum groups can arise from the same Hopf *-algebra.

Complex behaviors occur even in cocommutative cases, answering a question of Sołtan.

The Bohr compactification of  G is recovered via almost periodic function concepts.

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 So{\l}tan'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 So{\l}tan. 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 $\hat G$.