On the extension of unitary group isomorphisms of unital UHF-algebras

H. Dye showed that an isomorphism between the (discrete) unitary groups in two factors not of type In is implemented by a linear (or a conjugate linear)∗-isomorphism of the factors. If φ is an isomorphism between the unitary groups of two unital C∗-algebras, it induces a bijective map θφ between the sets of projections. For certain UHF-algebras, we construct an automorphism φ of their unitary group, such that θφ does not preserve the orthogonality of projections. For a large class of unital finite C∗-algebras, we show that θφ is always an orthoisomorphism. If φ is a continuous automorphism of the unitary group of a UHF-algebra A, we show that φ is implemented by a linear or a conjugate linear∗-automorphism of A.