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Extensions of stable \(C^*\)-algebras. (English) Zbl 1007.46045
From the Brown-Douglas-Fillmore theory it is known [L. G. Brown, R. G. Douglas and P. A. Fillmore, Lect. Notes Math. 345, 58-128 (1973; Zbl 0277.46053)] that for every extension \[ 0 \to K \to A \to B \to 0 \] 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 \[ 0 \to C(Z) \otimes K \to A \to K \to 0 \] has been found for some (infinite dimensional) compact Hausdorff space \(Z\) such that \(A\) is not stable.

46L05 General theory of \(C^*\)-algebras
46L35 Classifications of \(C^*\)-algebras
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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