Uniqueness of extremal almost periodic states on the injective type III₁ factor

Let R_∞ denote the Araki--Woods factor -- the unique separable injective type III₁ factor. For extremal almost periodic states φ, ψ∈ (R_∞)_*, we show that if Δ_φ and Δ_ψ have the same point spectrum then ψ = φ∘ α for some α∈ Aut(R_∞). Consequently, the extremal almost periodic states on R_∞ are parameterized by countable dense subgroups of R₊, up to precomposition by automorphisms. As an application, we show that KMS states for generalized gauge actions on Cuntz algebras agree (up to an automorphism) with tensor products of Powers states on their von Neumann completions.