Show simple item record

dc.contributor.authorGardella, Eusebio
dc.contributor.authorLupini, Martino
dc.date.accessioned2017-05-04T09:22:23Z
dc.date.available2017-05-04T09:22:23Z
dc.date.issued2017-01-17
dc.identifier.urihttp://publications.mfo.de/handle/mfo/1280
dc.identifier.urihttps://arxiv.org/abs/1612.08217
dc.description2000 Mathematics Subject Classification. Primary 46L55, 54H05; Secondary 03E15, 37A55en_US
dc.descriptionOWLF 2016en_US
dc.description.abstractLet $\Lambda $ 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 $\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 $\Gamma$.en_US
dc.language.isoenen_US
dc.publisherMathematisches Forschungsinstitut Oberwolfachen_US
dc.relation.ispartofseriesOberwolfach Preprints;2017,01
dc.titleCocycle Superrigidity and Group Actions on Stably Finite C*-Algebrasen_US
dc.typePreprinten_US
dc.identifier.doi10.14760/OWP-2017-01
local.scientificprogramOWLF 2016en_US
local.series.idOWP-2017-01
local.subject.msc46
local.subject.msc54
local.subject.msc03
local.subject.msc37


Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record