Remarks on the Quantum Bohr Compactification

Daws, Matthew orcid iconORCID: 0000-0003-1707-4308 (2014) Remarks on the Quantum Bohr Compactification. Illinois Journal of Mathematics, 57 (4). pp. 1131-1171. ISSN 0019-2082

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Official URL: https://doi.org/10.1215/ijm/1417442565

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 of G^.


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