Free products of finite dimensional and other von Neumann algebras with respect to non-tracial states

The von Neumann algebra free product of arbitary finite dimensional von Neumann algebras with respect to arbitrary faithful states, at least one of which is not a trace, is found to be a type~III factor possibly direct sum a finite dimensional algebra. The free product state on the type~III factor is what we call an extremal almost periodic state, and has centralizer isomorphic to $L({\text{freeF}}_{\mathrm{\infty }})$. This allows further classification the type~III factor and provides another construction of full type~III${}_1$ factors having arbitrary $\text{Sd}$~invariant of Connes. The free products considered in this paper are not limited to free products of finite dimensional algebras, but can be of a quite general form.