On invariant states and the commutant of a group of quasi-free automorphisms of the car algebra

We study primary states of the CAR algebra which are left invariant under quasi-free automorphisms α_U corresponding to unitaries U of a von Neumann algebra $\text{M}$ on the one-particle Hilbert space, and show that they are quasi-free states ?_A corresponding to self-adjoint operators A in $\text{M}$′ with 0 ? A ? 1, under the assumption that $\text{M}$ does not contain any finite type Ifactor direct summands. Next we study automorphisms of the CAR algebra which commute with α_U for U in a von Neumann algebra $\text{M}$ and show that they are quasi-free automorphisms α_U with U in $\text{M}$′ under the same assumption on $\text{M}$ as above. Finally by using the latter result we obtain a generalization of a theorem of Hugenholtz and Kadison [3].