ON THE EXTENSION OF UNITARY GROUP AUTOMORPHISMS OF CUNTZ ALGEBRAS

H. Dye showed that an isomorphism between the (discrete) unitary groups in two factors not of type I-n is implemented by a linear (or a conjugate linear) *-isomorphism of the factors. If phi is an isomorphism between the unitary groups of two unital C*-algebras, it induces a bijective map theta(phi) between the sets of projections. In this paper, we prove that the induced map theta(phi) is an orthoisomorphism, for the Cuntz algebras and for simple unital purely infinite C*-algebras having 2-divisible K-0-groups. Following Dye's approach, we prove that if phi is an automorphism of u(O-n), 2 <= n <= infinity, then there exists a unique *-automorphism psi (linear and conjugate linear) on O-n, such that psi = phi on a certain subgroup of O-n which is generated by self-adjoint unitaries of the form 1 - 2P(i,j)(a).