H. Dye showed that an isomorphism between the (discrete) unitarygroups in two factors not of type In 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 θφ between the sets ofprojections. In this paper, we prove that the induced map θφ 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(On) , 2 ≤ n≤ ∞, then there exists a unique ∗ -automorphism ψ(linear and conjugate linear) on On, such that ψ= φ on a certain subgroup of Onwhich is generated by self-adjoint unitaries of the form 1 - 2 Pi,j(a).
- Cuntz algebra
- classification with unitary groups
ASJC Scopus subject areas
- General Mathematics