Deformation of \(C^\ast\)-algebras by cocycles on locally compact quantum groups
type: Item
instance of: scholarly article
title: Deformation of \(C^\ast\)-algebras by cocycles on locally compact quantum groups
zbMATH Open document ID: 1317.46052
published in: Advances in Mathematics
publication date: 2014-04-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.4897
review text: The authors give a method to deform \(C^*\)-algebras endowed with actions of locally compact quantum groups. The deformation data consist of a \(C^*\)-algebra \(A\) together with a (left) action of a locally compact quantum group \(G\) and a unitary \(2\)-cocycle \(\Omega\) on its dual \(\widehat{G}\). One of the main results in the paper states that under a certain regularity condition on the \(2\)-cocycle, the deformed \(C^*\)-algebra \(A_\Omega\) is stably isomorphic to a twisted bidual crossed product of the form \(\widehat{G}^{\mathrm{op}}\ltimes_\Omega G \ltimes A\). The regularity condition on \(\Omega\) means that \(C_0(G)_\Omega\ltimes G\) is canonically isomorphic to the \(C^*\)-algebra \(K(L^2(G))\) of compact operators on the Hilbert space \(L^2(G)\). This is related to regularity of \(G\) itself, but even if \(G\) is a locally compact \textit{group} it is not clear whether every \(2\)-cocycle on \(G\) is regular. In this case, the regularity is strongly related to properness and saturation (in \textit{M. A. Rieffel}'s sense [Prog. Math. 84, 141--182 (1991; Zbl 0749.22003)]) for the action of \(G\) on the deformed algebra \(C_0(G)_\Omega=C^*(\widehat{G},\Omega)\). Indeed, the authors leave this problem open and also the task to find more manageable conditions of verifying regularity in general. In the group case, the deformation process defined by the authors is related to previous deformation constructions, as studied by Rieffel, Kasprzak, \textit{P. Bieliavsky} and \textit{V. Gayral} [Mem. Am. Math. Soc. 1110 (2015; \url{doi:10.1090/memo/1115}), \url{arxiv:1109.3419}], but the exact relation is not clear in general and is also left as an open problem in the paper.
reviewed by: Alcides Buss
Mathematics Subject Classification ID: 46L65
zbMATH DE Number: 6285008
zbMATH Keywords: locally compact quantum groups
zbMATH Keywords: cocycles
zbMATH Keywords: Rieffel's deformation
author: Sergey Neshveyev
author: Lars Tuset
MaRDI profile type: Publication
OpenAlex ID: W2083474360
arXiv ID: 1301.4897
cites work: Unitaires multiplicatifs et dualité pour les produits croisés de $\mathrm{C}^*$-algèbres
cites work: Non-semi-regular quantum groups coming from number theory
cites work: Deformation of operator algebras by Borel cocycles
cites work: Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups
cites work: Deformation Quantization for Actions of Kählerian Lie Groups
cites work: Quasi-hopf algebra actions and smash products
cites work: Noncommutative manifolds, the instanton algebra and isospectral deformations
cites work: Galois objects and cocycle twisting for locally compact quantum groups
cites work: On a Morita equivalence between the duals of quantum \(SU(2)\) and quantum \(\widetilde E(2)\)
cites work: Measured Quantum Groupoids in action
cites work: Twisting and Rieffel's deformation of locally compact quantum groups: deformation of the Haar measure
cites work: Noncommutative principal torus bundles via parametrised strict deformation quantization
cites work: Rieffel deformation via crossed products
cites work: Rieffel deformation of group coactions
cites work: Locally compact quantum groups.
cites work: Locally compact quantum groups in the von Neumann algebraic setting
cites work: Q4028311
cites work: Twisted dual-group algebras: Equivariant deformations of \(C_ 0(G)\)
cites work: Equivariant deformations of homogeneous spaces
cites work: Quasi-\(*\) structure on \(q\)-Poincaré algebras
cites work: Twisting in group algebras of finite groups
cites work: Smooth crossed products of Rieffel's deformations
cites work: ON SECOND COHOMOLOGY OF DUALS OF COMPACT GROUPS
cites work: A new quantum deformation of `\(ax+b\)' group
cites work: Q3973594
cites work: Deformation quantization for actions of 𝑅^{𝑑}
cites work: Integrable and proper actions on \(C^*\)-algebras, and square-integrable representations of groups
cites work: Twists and spectral triples for isospectral deformations
cites work: On the quantum `\(ax + b\)' group
cites work: A new approach to induction and imprimitivity results
cites work: Extensions of locally compact quantum groups and the bicrossed product construction.
cites work: Quantum symmetry groups of noncommutative spheres
cites work: Deformations of compact quantum groups via Rieffel's quantization
cites work: Rieffel type discrete deformation of finite quantum groups
cites work: Ergodic actions of compact groups on operator algebras. I: General theory
cites work: QUANTUM 'ax + b' GROUP
cites work: Produits croises d'une \(C^ *\)-algèbre par un groupe d'automorphismes
DOI: 10.1016/J.AIM.2013.12.025
description: scientific article