Homogeneity of the Pure State Space of the Cuntz Algebra

If ω1, ω2 are two pure gauge-invariant states of the Cuntz algebra Od, we show that there is an automorphism α of Od such that ω1=ω2∘α. If ω is a general pure state on Od and ϕ0 is a given Cuntz state, we show that there exists an endomorphism α of Od such that ϕ0=ω∘α.