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Model theory and Rokhlin dimension for compact quantum group actions

Gardella, Eusebio and Kalantar, Mehrdad and Lupini, Martino (2019) Model theory and Rokhlin dimension for compact quantum group actions. Journal of Noncommutative Geometry, 13 (2). pp. 711-767. ISSN 1661-6952 .

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We show that, for a given compact or discrete quantum group G, the class of actions of G on C*-algebras is first-order axiomatizable in the logic for metric structures. As an application, we extend the notion of Rokhlin property for G-C*-algebra, introduced by Barlak, Szabó, and Voigt in the case when G is second countable and coexact, to an arbitrary compact quantum group G. All the the preservations and rigidity results for Rokhlin actions of second countable coexact compact quantum groups obtained by Barlak, Szabó, and Voigt are shown to hold in this general context. As a further application, we extend the notion of equivariant order zero dimension for equivariant *-homomorphisms, introduced in the classical setting by the first and third authors, to actions of compact quantum groups. This allows us to define the Rokhlin dimension of an action of a compact quantum group on a C*-algebra, recovering the Rokhlin property as Rokhlin dimension zero. We conclude by establishing a preservation result for finite nuclear dimension and finite decomposition rank when passing to fixed point algebras and crossed products by compact quantum group actions with finite Rokhlin dimension.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Lupini, Martino0000-0003-1588-7057
Alternate Title:Rokhlin dimension for compact quantum group actions
Additional Information:© 2019 European Mathematical Society. Received 06 August, 2017. Published online: 2019-07-25. This work was initiated during a visit of E.G. and M.L. to the Mathematisches Forschungsinstitut Oberwolfach in August 2016, supported by an Oberwolfach Leibnitz Fellowship of M.L. Parts of this work were carried out during a visit of E.G. and M.K. to the California Institute of Technology in January 2017, and during a visit of E.G. and M.L. to the Centre de Recerca Matemàtica in March 2017 in occasion of the Intensive Research Programme on Operator Algebras. The authors gratefully acknowledge the hospitality and the financial support of all these institutions. E.G. was partially funded by SFB 878 Groups, Geometry and Actions, and by a postdoctoral fellowship from the Humboldt Foundation. M.K. was partially supported by the NSF Grant DMS-1700259. M.L. was partially supported by the NSF Grant DMS-1600186. This work is part of the project supported by the grant H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS.
Funding AgencyGrant Number
Oberwolfach Leibnitz FellowshipUNSPECIFIED
Centre de Recerca MatemàticaUNSPECIFIED
Deutsche Forschungsgemeinschaft (DFG)SFB 878
Alexander von Humboldt FoundationUNSPECIFIED
European UnionH2020-MSCA-RISE-2015-691246
Subject Keywords:C*-algebra, compact quantum group, quantum group action, Rokhlin property, Rokhlin dimension, logic for metric structures, axiomatizable class, positive existential embedding
Issue or Number:2
Classification Code:Mathematics Subject Classification (2010). 20G42, 46L55, 54H05; 03E15, 37A55
Record Number:CaltechAUTHORS:20180410-142028337
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Official Citation:Gardella Eusebio, Kalantar Mehrdad, Lupini Martino: Model theory and Rokhlin dimension for compact quantum group actions. J. Noncommut. Geom. 13 (2019), 711-767. doi: 10.4171/JNCG/337
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85731
Deposited By: Tony Diaz
Deposited On:11 Apr 2018 16:25
Last Modified:09 Aug 2019 21:00

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