On some free products of von Neumann algebras which are free Araki-Woods factors

We prove that certain free products of factors of type $\mathrm{I}$ and other von Neumann algebras with respect to nontracial, almost periodic states are almost periodic free Araki-Woods factors. In particular, they have the free absorption property and Connes’ Sd invariant completely classifies these free products. For example, for $\lambda ,\mu \in ]0,1[$, we show that $$(M_2(\text{C}),{\omega }_{\lambda })\ast (M_2(\text{C}),{\omega }_{\mu })$$ is isomorphic to the free Araki-Woods factor whose Sd invariant is the subgroup of ${\mathbb{R}}_{+}^{\ast }$ generated by $\lambda$ and $\mu$. Our proofs are based on algebraic techniques and amalgamated free products. These results give some answers to questions of Dykema and Shlyakhtenko.