# Journal of Operator Theory

Volume 76, Issue 2, Fall 2016  pp. 249-269.

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

Authors:  Thierry Giordano (1) and Adam Sierakowski (2)
Author institution:(1) Department of Mathematics and Statistics, University of Ottawa, Ottawa, K1N 6N5, Canada
(2) School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, 2522, Australia

Summary: In 1955 Dye proved that two von Neumann factors not of type I$_{2n}$ are isomorphic if and only if their unitary groups are isomorphic as abstract groups. We consider an analogue for $C^*$-algebras and show that the topological general linear group is a classifying invariant for simple unital AH-algebras of slow dimension growth and of real rank zero, and that the abstract general linear group is a classifying invariant for unital Kirchberg algebras in the UCT class.

DOI: http://dx.doi.org/10.7900/jot.2015may27.2112
Keywords: operator algebras, classification, general linear group