Das, B and Daws, M (2014) Quantum Eberlein compactifications and invariant means. arXiv. (Unpublished)
Abstract
We propose a definition of a "$C^*$-Eberlein" algebra, which is a weak form of a $C^*$-bialgebra with a sort of "unitary generator". Our definition is motivated to ensure that commutative examples arise exactly from semigroups of contractions on a Hilbert space, as extensively studied recently by Spronk and Stokke. The terminology arises as the Eberlein algebra, the uniform closure of the Fourier-Stieltjes algebra $B(G)$, has character space $G^{\mathcal E}$, which is the semigroup compactification given by considering all semigroups of contractions on a Hilbert space which contain a dense homomorphic image of $G$. We carry out a similar construction for locally compact quantum groups, leading to a maximal $C^*$-Eberlein compactification. We show that $C^*$-Eberlein algebras always admit invariant means, and we apply this to prove various "splitting" results, showing how the $C^*$-Eberlein compactification splits as the quantum Bohr compactification and elements which are annihilated by the mean. This holds for matrix coefficients, but for Kac algebras, we show it also holds at the algebra level, generalising (in a semigroup-free way) results of Godement.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2014 The Authors. Imported from arXiv |
Keywords: | Math.FA; math.OA; 43A07, 43A60, 46L89, 47D03 |
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 10:51 |
Last Modified: | 23 Jan 2018 23:56 |
Status: | Unpublished |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:83382 |