Free products of finite-dimensional and other von Neumann algebras in terms of free Araki–Woods factors

We show that any free product of finite-dimensional von Neumann algebras equipped with non-tracial states is isomorphic to a Araki–Woods factor its quasi-free state possibly direct sum algebra. This gives complete answer questions posed by Dykema in [5] and Shlyakhtenko [10], which had been partially answered Houdayer [7] Ueda [16]. also extend this suitable infinite-dimensional almost periodic states.