The unitary group as an invariant of a simple unital C*-algebra

Abstract

In 1954, H. Dye proved that the unitary groups of von Neumann factors not of type I2n determine the algebraic type of factors. Using Dye's result, M. Broise showed that any isomorphism between the unitary groups of two von Neumann factors not of type In is implemented by a linear or a conjugate linear *-isomorphism between the factors. Using Dye's approach, A. Booth proved that two simple unital AF-algebras are isomorphic if and only if their unitary groups are (algebraically) isomorphic. In the first part of this thesis, we extend Booth's result to a larger class of amenable unital C*-algebras. If ϕ is an isomorphism between the unitary groups of two unital C*-algebras, it induces a bijective map &thgr;ϕ between the sets of projections of the algebras. For some UHF-algebras, we construct an automorphism ϕ of their unitary group, such that &thgr;ϕ does not preserve the orthogonality of projections. For a large class of unital C*-algebras, we show that &thgr;ϕ is always an orthoisomorphism. This class includes in particular the Cuntz algebras On , 2 ≤ n ≤ infinity, 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.