2,179 research outputs found
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
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
Turbulence, representations, and trace-preserving actions
We establish criteria for turbulence in certain spaces of C*-algebra
representations and apply this to the problem of nonclassifiability by
countable structures for group actions on a standard atomless probability space
(X,\mu) and on the hyperfinite II_1 factor R. We also prove that the conjugacy
action on the space of free actions of a countably infinite amenable group on R
is turbulent, and that the conjugacy action on the space of ergodic
measure-preserving flows on (X,\mu) is generically turbulent.Comment: 27 page
Linearization of Hyperbolic Finite-Time Processes
We adapt the notion of processes to introduce an abstract framework for
dynamics in finite time, i.e.\ on compact time sets. For linear finite-time
processes a notion of hyperbolicity namely exponential monotonicity dichotomy
(EMD) is introduced, thereby generalizing and unifying several existing
approaches. We present a spectral theory for linear processes in a coherent
way, based only on a logarithmic difference quotient, prove robustness of EMD
with respect to a suitable (semi-)metric and provide exact perturbation bounds.
Furthermore, we give a complete description of the local geometry around
hyperbolic trajectories, including a direct and intrinsic proof of finite-time
analogues of the local (un)stable manifold theorem and theorem of linearized
asymptotic stability. As an application, we discuss our results for ordinary
differential equations on a compact time-interval.Comment: 32 page
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