Factor state extensions of type III

Let B be a C^*-subalgebra of A and θ{symbol} a factor state on B. Due to the independent work of S. Popa and R. Longo the existence of a factor state extension of θ{symbol} to A is assured in case of a separable B. However, the type of the extending factor state can't be determined according to their highly nonconstructive procedures. In this article we consider a type II₁, factor state θ{symbol} on the Choi algebra embedded as a C^*-subalgebra of the Cuntz algebra O₂. Type III_λ factor state extensions of θ{symbol} are constructed for λ = 1 and many λ's between 0 and 1.