On stability of \(C^*\)-algebras (Q1266256)

A \(C^\ast\)-algebra is called stable if \(A\) is isomorphic to \(A\otimes {\mathcal K}\), where \(\mathcal K\) is the algebra of compact operators on a separable Hilbert space. The main result of the paper says that a \(\sigma\)-unital algebra \(A\) is stable if and only if for each positive element \(a\in A\) and each \(\varepsilon>0\) there exists a positive element \(b\in A\) such that \(| | ab| | <\varepsilon\) and \(a\sim b,\) meaning that \(x^\ast x=a, xx^\ast=b\) for some \(x\in A\). Also, \(A\) is stable if and only if there is a sequence \((E_n)\) of mutually orthogonal and equivalent projections in \(M(A)\), the multiplier algebra of \(A\), such that \(\sum E_n=1\) in the strict topology. Moreover, it is shown that a \(C^\ast\)-algebra with a countable approximate unit consisting of projections is stable if and only if for each projection \(p\in A\) there is a projection \(q\in A\) such that \(p\sim q\) and \(pq=0\). By using these characterizations the authors show that, e.g., any inductive limit of a sequence of \(\sigma\)-unital stable \(C^\ast\)-algebras is stable, and that the crossed product of a \(\sigma\)-unital stable \(C^\ast\)-algebra by a discrete group is stable.