On Stability of C*-algebras

Let A be a oe-unital C -algebra, i.e. A admits a countable approximate unit. It is proved that A is stable, i.e. A is isomorphic to A\Omega K where K is the algebra of compact operators on a separable Hilbert space, if and only if for each positive element a 2 A and each " ? 0 there exists a positive element b 2 A such that kabk ! " and x x = a, xx = b for some x in A. Using this characterization it is proved among other things that the inductive limit of any sequence of oe-unital stable C -algebras is stable, and that the crossed product of a oe-unital stable C -algebra by a discrete group is again stable. 1 Introduction One can characterize stable AF-algebras as being precisely those AF-algebras that do not admit a bounded densely defined trace. This can be seen by using the classification of AF-algebras by their ordered K 0 -group (see also Section 5). One motivation for this paper is if a similar strong characterization of stable C -algebras might hold in g..

On Stability of C*-algebras

Let A be a oe-unital C*-algebra, i.e. A admits a countable approximate unit. It is proved that A is stable, i.e. A is isomorphic to A\Omega K where K is the algebra of compact operators on a separable Hilbert space, if and only if for each positive element a 2 A and each " ? 0 there exists a positive element b 2 A such that kabk ! " and x x = a, xx = b for some x in A. Using this characterization it is proved among other things that the inductive limit of any sequence of oe-unital stable C -algebras is stable, and that the crossed product of a oe-unital stable C -algebra by a discrete group is again stable