On the Classification of C*- Algebras Using Unitary Groups

In 1955, Dye proved that the discrete unitary group in a factor determines the algebraic type of the factor. Using Dye's approach, we prove similar results to a larger class of amenable unital C-algebras including simple unital AH-algebras (of SDG) with real rank zero. 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 of the algebras. For some 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 C-algebras, we show that θφ is always an orthoisomorphism. This class includes in particular the Cuntz algebras On, 2  n  1, and the simple unital AF-algebras having 2-divisible K0-group. 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.