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Classification of extensions of classifiable \(C^{*}\)-algebras. (English) Zbl 1207.46055
Authors’ abstract: For a certain class of extensions \({\mathfrak e}: 0\rightarrow B \rightarrow E \rightarrow A \rightarrow 0\) of \(C^*\)-algebras in which \(B\) and \(A\) belong to classifiable classes of \(C^*\)-algebras, we show that the functor which sends \({\mathfrak e}\) to its associated six term exact sequence in \(K\)-theory and the positive cones of \(K_0 (B)\) and \(K_0 (A)\) is a classification functor. We give two independent applications addressing the classification of a class of \(C^*\)-algebras arising from substitutional shift spaces on one hand and of graph algebras on the other. The former application leads to the answer of a question of Carlsen and the first author concerning the completeness of stabilized Matsumoto algebras as an invariant of flow equivalence. The latter leads to the first classification result for nonsimple graph \(C^*\)-algebras.

MSC:
46L35 Classifications of \(C^*\)-algebras
19K14 \(K_0\) as an ordered group, traces
19K33 Ext and \(K\)-homology
19K35 Kasparov theory (\(KK\)-theory)
37B10 Symbolic dynamics
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
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