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

摘要

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. For certain UHF-algebras, we construct an automorphism phi of their unitary group, such that theta(phi) does not preserve the orthogonality of projections. For a large class of unital finite C*-algebras, we show that theta(phi) is always an orthoisomorphism. If phi is a continuous automorphism of the unitary group of a UHF-algebra A, we show that. is implemented by a linear or a conjugate linear *-automorphism of A.