CaltechAUTHORS
  A Caltech Library Service

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. http://resolver.caltech.edu/CaltechAUTHORS:20171219-145613907

[img] PDF - Accepted Version
See Usage Policy.

475Kb
[img] PDF - Submitted Version
See Usage Policy.

445Kb

Use this Persistent URL to link to this item: http://resolver.caltech.edu/CaltechAUTHORS:20171219-145613907

Abstract

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
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.jfa.2017.12.005DOIArticle
http://www.sciencedirect.com/science/article/pii/S0022123617304585PublisherArticle
https://arxiv.org/abs/1612.08217arXivDiscussion Paper
ORCID:
AuthorORCID
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.
Funders:
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)SFB 878
Alexander von Humboldt FoundationUNSPECIFIED
NSFDMS-1600186
Mathematisches Forschungsinstitut OberwolfachUNSPECIFIED
Subject Keywords:Property (T); 1-Cohomology; Profinite group; Rokhlin property; Model action; Borel complexity; Conjugacy; Cocycle conjugacy; Complete analytic set
Classification Code:2000 Mathematics Subject Classification: Primary 46L55, 54H05; Secondary 03E15, 37A55
Record Number:CaltechAUTHORS:20171219-145613907
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20171219-145613907
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, https://doi.org/10.1016/j.jfa.2017.12.005. (http://www.sciencedirect.com/science/article/pii/S0022123617304585)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:83970
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:19 Dec 2017 23:24
Last Modified:24 May 2018 16:16

Repository Staff Only: item control page