The general linear group as a complete invariant for $C^*$-algebras

Abstract

In 1955 Dye proved that two von Neumann factors not of type I2n are isomorphic (via a linear or a conjugate linear u -isomorphism) if and only if their unitary groups are isomorphic as abstract groups. We consider an analogue for C u -algebras. We show that the topological gen- eral linear group is a classifying invariant for simple, unital AH-algebras of slow dimension growth and of real rank zero, and the abstract gen- eral linear group is a classifying invariant for unital Kirchberg algebras in UCT.