On fully ∗-extendable automorphisms of the unitary group of uhf-algebras

If ϕ is an automorphism of the unitary group of UHF-algebras and the induced map θ_ϕ on the projections is an orthoisomorphism, then there exists a ∗-automorphism ψ such that ψ = ϕ on a subgroup containing the self-adjoint unitaries in the finite dimensional subalgebras. Indeed, if ϕ is continuous, then ϕ is implemented by a ∗-automorphism of the UHF-algebras. In this paper, we construct automorphisms of the unitary groups of certain UHF-algebras such that no ∗-automorphism coincides with ϕ on the unitary group.