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Actions of rigid groups on UHF-algebras

Gardella, Eusebio and Lupini, Martino (2018) Actions of rigid groups on UHF-algebras. Journal of Functional Analysis, 275 (2). pp. 381-421. ISSN 0022-1236. doi:10.1016/j.jfa.2017.12.005.

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Let Λ be a countably infinite property (T) group, and let D be UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of Λ on D. The proof consists in assigning, to any second countable abelian pro-p group G, a strongly outer action of Λ on D whose (weak) cocycle conjugacy class completely remembers the group G. The group G is reconstructed from the action through its (weak) 1-cohomology set endowed with a canonical pairing function. Our construction also shows the following stronger statement: the relations of conjugacy, cocycle conjugacy, and weak cocycle conjugacy of strongly outer actions of Λ on D are complete analytic sets, and in particular not Borel. The same conclusions hold more generally when Λ is only assumed to contain an infinite subgroup with relative property (T), and for actions on (not necessarily simple) separable, nuclear, UHF-absorbing, self-absorbing C*-algebras with at least one trace. Finally, we use the techniques of this paper to construct outer actions on R with prescribed cohomology. Precisely, for every infinite property (T) group Λ, and for every countable abelian group Γ, we construct an outer action of Λ on R whose 1-cohomology is isomorphic to Γ.

Item Type:Article
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URLURL TypeDescription Paper
Lupini, Martino0000-0003-1588-7057
Additional Information:© 2017 Elsevier Inc. Received 28 August 2017, Accepted 13 December 2017, Available online 19 December 2017. The first-named author was partially funded by SFB 878Groups, Geometry and Actions, and by a postdoctoral fellowship from the Humboldt Foundation. The second-named author was partially supported by the NSF Grant DMS-1600186. This work was initiated during a visit of the authors at the Mathematisches Forschungsinstitut Oberwolfach in August 2016 supported by an Oberwolfach Leibnitz Fellowship. The authors gratefully acknowledge the hospitality of the Institute.
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)SFB 878
Alexander von Humboldt FoundationUNSPECIFIED
Mathematisches Forschungsinstitut OberwolfachUNSPECIFIED
Subject Keywords:Property (T); 1-Cohomology; Profinite group; Rokhlin property; Model action; Borel complexity; Conjugacy; Cocycle conjugacy; Complete analytic set
Issue or Number:2
Classification Code:2000 Mathematics Subject Classification: Primary 46L55, 54H05; Secondary 03E15, 37A55
Record Number:CaltechAUTHORS:20171219-145613907
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Official Citation:Eusebio Gardella, Martino Lupini, Actions of rigid groups on UHF-algebras, Journal of Functional Analysis, Volume 275, Issue 2, 2018, Pages 381-421, ISSN 0022-1236, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:83970
Deposited By: Tony Diaz
Deposited On:19 Dec 2017 23:24
Last Modified:15 Nov 2021 20:15

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