191 research outputs found
Borel reducibility and classification of von Neumann algebras
We announce some new results regarding the classification problem for
separable von Neumann algebras. Our results are obtained by applying the notion
of Borel reducibility and Hjorth's theory of turbulence to the isomorphism
relation for separable von Neumann algebras
Turbulence and Araki-Woods factors
Using Baire category techniques we prove that Araki-Woods factors are not
classifiable by countable structures. As a result, we obtain a far reaching
strengthening as well as a new proof of the well-known theorem of Woods that
the isomorphism problem for ITPFI factors is not smooth. We derive as a
consequence that the odometer actions of Z that preserve the measure class of a
finite non-atomic product measure are not classifiable up to orbit equivalence
by countable structures.Comment: 16 page
Logic and operator algebras
The most recent wave of applications of logic to operator algebras is a young
and rapidly developing field. This is a snapshot of the current state of the
art.Comment: A minor chang
The classification problem for automorphisms of C*-algebras
We present an overview of the recent developments in the study of the
classification problem for automorphisms of C*-algebras from the perspective of
Borel complexity theory.Comment: 21 page
Games orbits play and obstructions to Borel reducibility
We introduce a new, game-theoretic approach to anti-classification results
for orbit equivalence relations. Within this framework, we give a short
conceptual proof of Hjorth's turbulence theorem. We also introduce a new
dynamical criterion providing an obstruction to classification by orbits of CLI
groups. We apply this criterion to the relation of equality of countable sets
of reals, and the relations of unitary conjugacy of unitary and selfadjoint
operators on the separable infinite-dimensional Hilbert space.Comment: 13 pages. Final version, to appear in Groups, Geometry, and Dynamic
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