Unitary groups as a complete invariant

Dye proved that the discrete unitary group in a factor determines the algebraic type of the factor. We show that if the unitary groups of two simple unital AH-algebras of slow dimension growth and of real rank zero are isomorphic as abstract groups, then their K ₀-ordered groups are isomorphic. Also, using Gong and Dadarlat's classification theorem, we prove that such C ^*-algebras are isomorphic if and only if their unitary groups are isomorphic as topological groups. For simple, unital purely infinite C ^*-algebras, we show that two unital Kirchberg algebras are *-isomorphic if and only if their unitary groups are isomorphic as abstract groups.