On Stability ofC*-Algebras

LetAbe aσ-unitalC*-algebra, i.e.,Aadmits a countable approximate unit. It is proved thatAis stable, i.e.,Ais isomorphic toA⊗K where K is the algebra of compact operators on a separable Hilbert space, if and only if for each positive elementa∈Aand eachε>0 there exists a positive elementb∈Asuch that ‖ab‖<εandx*x=a,xx*=bfor somexinA. Using this characterization it is proved among other things that the inductive limit of any sequence ofσ-unital stableC*-algebras is stable, and that the crossed product of aσ-unital stableC*-algebra by a discrete group is again stable.