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 K0-ordered groups are isomorphic. Also, using Gong and Dadarlatʼs 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.