Extensions of stable \(C^*\)-algebras (Q5946780)

From the Brown-Douglas-Fillmore theory it is known [\textit{L. G. Brown, R. G. Douglas} and \textit{P. A. Fillmore}, Lect. Notes Math. 345, 58-128 (1973; Zbl 0277.46053)] that for every extension NEWLINE\[NEWLINE0 \to K \to A \to B \to 0NEWLINE\]NEWLINE of separable \(C^*\)-algebras one has \(A\) is stable if and only if \(B\) is stable. A natural question arises whether every extension of two (separable) stable \(C^*\)-algebras is stable. The author gives a negative answer to this question. More precisely an extension NEWLINE\[NEWLINE0 \to C(Z) \otimes K \to A \to K \to 0NEWLINE\]NEWLINE has been found for some (infinite dimensional) compact Hausdorff space \(Z\) such that \(A\) is not stable.