Free Quasi-free States

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 freeprobability 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.