Actions of xerox type on Araki-Woods factors and their fixed point von Neumann algebras

Abstract

The von Neumann algebras studied in this thesis arise as GNS-representation of fixed point algebras under a certain type of group actions, called xerox actions, on A = ⊗Mk( C ), the kinfinity-UHF algebra. The xerox actions are induced by unitary representations of compact groups on Mk( C ). The states used to perform the GNS representation are restrictions of diagonal product states on kinfinity-UHF algebras. First, we analyze xerox actions induced by diagonal representations of compact groups. In this case the von Neumann algebras studied here can be seen as fixed point algebras under xerox actions of compact groups on Araki-Woods factors. We obtain necessary and sufficient conditions for such von Neumann algebras to be factors and we determine their type. We also study the GNS representation of the fixed point algebra under the xerox action induced by non-diagonal representations of a compact groups G. We show that the von Neumann algebra obtained in this way can be identified up to isomorphism with the GNS representation of a fixed point algebra under a xerox action induced by a diagonal representation of a closed subgroup of G. Any von Neumann algebra, N, studied here can be realized as W*(X, mu, R ), the von Neumann algebra associated to an equivalence relation R on a measured space (X, mu). We give sufficient conditions for N to be isomorphic to an Araki-Woods factor. To prove this we show that the associated flow of R is approximately transitive. We study also when the equivalence relation R , that corresponds to N has Krieger's property A, and we prove that there exist equivalence relations which have property A but their associated flow is not approximately transitive.