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Free quasi-free states. (English) Zbl 0882.46026
Summary: To a real Hilbert space and a one-parameter group of orthogonal transformations we associate a $$C^*$$-algebra which admits a free quasi-free state. This construction is a free-probability analog of the construction of quasi-free states on the CAR and CCR algebras. We show that under certain conditions, our $$C^*$$-algebras are simple, and the free quasi-free states are unique.
The corresponding von Neumann algebras obtained via the GNS construction are free analogs of the Araki-Woods factors. Such von Neumann algebras can be decomposed into free products of other von Neumann algebras. For non-trivial one-parameter groups, these von Neumann algebras are type III factors. In the case the one-parameter group is nontrivial and almost-periodic, we show that Connes’ Sd invariant completely classifies these algebras.

##### MSC:
 46L30 States of selfadjoint operator algebras 46L35 Classifications of $$C^*$$-algebras 46L60 Applications of selfadjoint operator algebras to physics 46L55 Noncommutative dynamical systems
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