On the classification of nuclear C*-algebras
On the classification of nuclear C*-algebras
- Authors: Marius Dadarlat and Søren Eilers.
- Date: preliminary version June 1998, first version August 1998,
second version May 1999.
- Status: Appeared in Proceedings of the London
Mathematical Society, 85 (2002), 168-210.
- Pages: 43.
- Abstract: The mid-seventies' works on C*-algebras of Brown-Douglas-Fillmore and Elliott both contained uniqueness and existence results
in a now standard sense. These papers served as keystones for two
separate theories -- KK-theory and the classification program -- which
for many years parted ways with only moderate interaction. But recent
years have seen a fruitful interaction which has been one of the main
engines behind rapid progress in the classification program.
In the present paper we take this interaction even further.
We prove general existence and uniqueness results using KK-theory and
a concept of quasidiagonality for representations. These results are
employed to obtain new classification results for certain classes of
quasidiagonal C*-algebras introduced by H. Lin. An important novel
feature of these classes is that they are defined by a certain local
approximation property, rather than by an inductive limit construction.
Our existence and uniqueness results are in the spirit of
classical Ext-theory. The main complication overcome in the paper is
to control the stabilization which is necessary when one works with
finite C*-algebras. In the infinite case, where programs of this type
have already been successfully carried out, stabilization is
unnecessary. Yet, our methods are sufficiently versatile to allow us
to reprove, from a handful of basic results, the classification of
purely infinite nuclear C*-algebras of Kirchberg and Phillips.
Indeed, it is our hope that this can be the starting point of
a unified approach to classification of nuclear C*-algebras.
- Reference list: HTML,
postscript.
- Remarks:
- Access opportunities:
- From CUP (pdf format) here.
eilers@math.ku.dk/August
1998