Brannan, M, Daws, M and Samei, E (2013) Completely bounded representations of convolution algebras of locally compact quantum groups. Munster Journal of Mathematics, 6 (2). pp. 445-482. ISSN 1867-5778
Abstract
Given a locally compact quantum group G, we study the structure of completely bounded homomorphisms π:L1(G)→B(H), and the question of when they are similar to ∗-homomorphisms. By analogy with the cocommutative case (representations of the Fourier algebra A(G)), we are led to consider the associated map π∗:L1♯(G)→B(H) given by π∗(ω)=π(ω♯)∗. We show that the corepresentation Vπ of L∞(G) associated to π is invertible if and only if both π and π∗ are completely bounded. Moreover, we show that the co-efficient operators of such representations give rise to completely bounded multipliers of the dual convolution algebra $L^1(\hat \mathbb G)$. An application of these results is that any (co)isometric corepresentation is automatically unitary. An averaging argument then shows that when G is amenable, π is similar to a *-homomorphism if and only if π∗ is completely bounded. For compact Kac algebras, and for certain cases of A(G), we show that any completely bounded homomorphism π is similar to a *-homomorphism, without further assumption on π∗. Using free product techniques, we construct new examples of compact quantum groups G such that L1(G) admits bounded, but not completely bounded, representations.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Environment (Leeds) > School of Geography (Leeds) > Centre for Spatial Analysis & Policy (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 26 May 2017 11:50 |
Last Modified: | 26 May 2017 11:50 |
Published Version: | https://www.uni-muenster.de/FB10/mjm/vol_6/mjm_vol... |
Status: | Published |
Publisher: | Mathematical Institutes Münster, Universität Münster |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:116938 |