On the extension of unitary group automorphisms of Cuntz algebras

H. Dye showed that an isomorphism between the (discrete) unitarygroups in two factors not of type I_n is implemented by a linear (or a conjugatelinear) ∗ -isomorphism of the factors. If φ is an isomorphism between the unitarygroups of two unital C^∗-algebras, it induces a bijective map θ_φ between the sets ofprojections. In this paper, we prove that the induced map θ_φ is an orthoisomorphism,for the Cuntz algebras and for simple unital purely infinite C^∗-algebrashaving 2-divisible K-groups. Following Dye’s approach, we prove that if φ is anautomorphism of U(O_n) , 2 ≤ n≤ ∞, then there exists a unique ∗ -automorphism ψ(linear and conjugate linear) on O_n, such that ψ= φ on a certain subgroup of O_nwhich is generated by self-adjoint unitaries of the form 1 - 2 P_i,j(a).