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Winter 2013 Remarks on the quantum Bohr compactification
Matthew Daws
Illinois J. Math. 57(4): 1131-1171 (Winter 2013). DOI: 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 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 $\hat{G}$.

Citation

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Matthew Daws. "Remarks on the quantum Bohr compactification." Illinois J. Math. 57 (4) 1131 - 1171, Winter 2013. https://doi.org/10.1215/ijm/1417442565

Information

Published: Winter 2013
First available in Project Euclid: 1 December 2014

zbMATH: 1305.43006
MathSciNet: MR3285870
Digital Object Identifier: 10.1215/ijm/1417442565

Subjects:
Primary: 43A30, 43A60, 46L89
Secondary: 22D25, 43A20, 43A95, 47L25

Rights: Copyright © 2013 University of Illinois at Urbana-Champaign

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Vol.57 • No. 4 • Winter 2013
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