An abstract Voiculescu–Brown–Douglas–Fillmore absorption theorem

George A. Elliott and Dan Kucerovsky

Vol. 198 (2001), No. 2, 385–409

DOI: 10.2140/pjm.2001.198.385

Abstract

A common generalization is given of what are often referred to as the Weyl–von Neumann theorems of Voiculescu, Kasparov, Kirchberg, and, more recently, Lin. (These in turn extend a result of Brown, Douglas, and Fillmore.)

More precisely, an intrinsic characterization is obtained of those extensions of one separable C^∗-algebra by another—the first, i.e., the ideal, assumed to be stable, so that Brown-Douglas-Fillmore addition of extensions can be carried out—which are absorbing in a certain natural sense related to this addition, a sense which reduces to that considered by earlier authors if either the ideal or the quotient is nuclear. The specific absorption theorems referred to above can be deduced from this characterization.

Milestones

Received: 30 March 1999

Revised: 1 March 2000

Published: 1 April 2001