Rigidity of free product von Neumann algebras

Let $I$ be any nonempty set and $(M_i,{\phi }_i{)}_{i\in I}$ any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class ${\mathcal{C}}_{\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ of (possibly type III) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing a Cartan subalgebra. For the free product $(M, \varphi) = \ast_{i \in I} (M_i, \varphi_i)$, we show that the free product von Neumann algebra $M$ retains the cardinality $\mathrm{\mid }I\mathrm{\mid }$ and each nonamenable factor $M_i$ up to stably inner conjugacy, after permutation of the indices. Our main theorem unifies all previous Kurosh-type rigidity results for free product type II${}_1$ factors and is new for free product type III factors. It moreover provides new rigidity phenomena for type III factors.