dc.contributor.author Gardella, Eusebio dc.contributor.author Lupini, Martino dc.date.accessioned 2017-05-04T09:22:23Z dc.date.available 2017-05-04T09:22:23Z dc.date.issued 2017-01-17 dc.identifier.uri http://publications.mfo.de/handle/mfo/1280 dc.identifier.uri https://arxiv.org/abs/1612.08217 dc.description 2000 Mathematics Subject Classification. Primary 46L55, 54H05; Secondary 03E15, 37A55 en_US dc.description OWLF 2016 en_US dc.description.abstract Let $\Lambda$ be a countably infinite property (T) group, and let $D$ be en_US UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of $\Lambda$ on $D$. The proof consists in assigning, to any second countable abelian pro-$p$ group $G$, a strongly outer action of $\Lambda$ on $D$ whose (weak) cocycle conjugacy class completely remembers the group $G$. The group $G$ is reconstructed from the action via its (weak) 1-cohomology set endowed with a canonical pairing function. The key ingredient in this computation is Popa's cocycle superrigidity theorem for Bernoulli shifts on the hyperfinite II$_{1}$ factor $R$. Our construction also shows the following stronger statement: the relations of conjugacy, cocycle conjugacy, and weak cocycle conjugacy of strongly outer actions of $\Lambda$ on $D$ are complete analytic sets, and in particular not Borel. The same conclusions hold more generally when $\Lambda$ 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 $\Lambda$, and for every countable abelian group $\Gamma$, we construct an outer action of $\Lambda$ on $R$ whose 1-cohomology is isomorphic to $\Gamma$. dc.language.iso en en_US dc.publisher Mathematisches Forschungsinstitut Oberwolfach en_US dc.relation.ispartofseries Oberwolfach Preprints;2017,01 dc.title Cocycle Superrigidity and Group Actions on Stably Finite C*-Algebras en_US dc.type Preprint en_US dc.identifier.doi 10.14760/OWP-2017-01 local.scientificprogram OWLF 2016 en_US local.series.id OWP-2017-01 local.subject.msc 46 local.subject.msc 54 local.subject.msc 03 local.subject.msc 37
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