Daws, M (2013) Remarks on the quantum Bohr compactification. Illinois Journal of Mathematics, 4 (4). 1131 - 1171. ISSN 0019-2082
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 $\hat G$.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2013, University of Illinois at Urbana-Champaign. This is an author produced version of a paper published in Illinois Journal of Mathematics. |
Keywords: | math.FA; math.OA; Primary 43A60, 46L89, Secondary 22D25, 43A20, 43A30, 43A95, 47L25 |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 27 Mar 2015 10:45 |
Last Modified: | 31 Jan 2018 21:47 |
Status: | Published |
Publisher: | University of Illinois at Urbana-Champaign |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:83380 |