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.
