Browsing by Author "Lupini, Martino"

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    Boundary representations of operator spaces, and compact rectangular matrix convex sets
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Fuller, Adam H.; Hartz, Michael; Lupini, Martino
    We initiate the study of matrix convexity for operator spaces. We dene the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein-Milman and the bipolar theorems in this context. We deduce a canonical correspondence between compact rectangular matrix convex sets and operator spaces. We also introduce the notion of boundary representation for an operator space, and prove the natural analog of Arveson's conjecture: every operator space is completely normed by its boundary representations. This yields a canonical construction of the triple envelope of an operator space.
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    Cocycle superrigidity and group actions on stably finite C*-algebras
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Gardella, Eusebio; Lupini, Martino
    Let be a countably innite property (T) group, and let A be UHF-algebra of innite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of on A. The proof consists in assigning, to any second countable abelian pro-p group G, a strongly outer action of on A 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 hypernite II1 factor. Our construction also shows the following stronger statement: the relations of conjugacy, cocycle conjugacy, and weak cocycle conjugacy of strongly outer actions of on A are complete analytic sets, and in particular not Borel. The same conclusions hold more generally when is only assumed to contain an innite subgroup with relative property (T), and A is a (not necessarily simple) separable, nuclear, UHF-absorbing, self-absorbing C*-algebra with at least one trace.