Remarks on the Quantum Bohr Compactification

Abstract

The category of locally compact quantum groups can be described as either Hopf $\ast$-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${}^{\ast }$-algebraic compact quantum groups are produced, but the underlying Hopf $\ast$-algebras are always, canonically, the same. We show that a complicated range of behaviours, with C${}^{\ast }$-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 $\ast$-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 $\hat{G}$.

32 pages; some corrections and additions; to appear in Illinois Journal

Cited by in corpus (4)

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• Non-commutative separate continuity and weakly almost periodicity for Hopf von Neumann algebras
• Weak mixing for locally compact quantum groups
• On C*-completions of discrete quantum group rings