Classifying \(C^\ast\)-algebras with both finite and infinite subquotients. (English) Zbl 1294.46052
J. Funct. Anal. 265, No. 3, 449-468 (2013); corrigendum ibid. 270, No. 2, 854-859 (2016).
This paper is concerned with the classification of \(C^*\)-algebras \({\mathfrak E}\) over finite \(T_0\) topological spaces, assuming that the simple sub-quotients of \({\mathfrak E}\) are classifiable by \(K\)-theoretical invariants. An important step exhibits classes of extensions \(0\rightarrow {\mathfrak B} \rightarrow {\mathfrak E} \rightarrow {\mathfrak A} \rightarrow 0\) of \(C^*\)-algebras such that, if both \({\mathfrak A}\) and \({\mathfrak B}\) are classifiable by \(K\)-theoretical invariants, then the same holds for \({\mathfrak E}\). The main technical result, Theorem 4.6, considers the situation where \({\mathfrak E}\) is a \(C^*\)-algebra over a finite \(T_0\) topological space \(X\) with a non-trivial open subset \(U\) such that \({\mathfrak B}\) is a \(C^*\)-algebra over \(U\) and \({\mathfrak A}\) a \(C^*\)-algebra over \(Y=X\setminus U\). Assuming certain technical conditions on two full extensions \(0\rightarrow {\mathfrak B}_i \rightarrow {\mathfrak E}_i \rightarrow {\mathfrak A}_i \rightarrow 0\), \(i=1,2\), as above, it is proved that, if there exists an isomorphism \(\alpha:\text{FK}_X ({\mathfrak E}_1)\rightarrow \text{FK}_X ({\mathfrak E}_2)\) at the level of filtered \(K\)-theory which lifts to an invertible element in \(KK(X;{\mathfrak E}_1,{\mathfrak E}_2)\) and induces isomorphisms \(\alpha_U :\text{FK}_U^+ ({\mathfrak B}_1) \rightarrow \text{FK}_U^+ ({\mathfrak B}_2)\) and, respectively, \(\alpha_Y :\text{FK}_Y^+ ({\mathfrak A}_1) \rightarrow \text{FK}_Y^+ ({\mathfrak A}_2)\) at the level of filtered ordered \(K\)-theory, then the \(C^*\)-algebras \({\mathfrak E}_1\) and \({\mathfrak E}_2\) are isomorphic. The possibility of removing some of these technical assumptions is analysed. This extends an earlier work of the authors [Adv. Math. 222, No. 6, 2153–2172 (2009; Zbl 1207.46055)].
These results are then applied to show that a large class of graph \(C^*\)-algebras, associated to finite linear lattices with no more than one transition from finite to infinite sub-quotients, are classified up to stable isomorphisms by their filtered ordered \(K\)-theory.
These results are then applied to show that a large class of graph \(C^*\)-algebras, associated to finite linear lattices with no more than one transition from finite to infinite sub-quotients, are classified up to stable isomorphisms by their filtered ordered \(K\)-theory.
Reviewer: Florin P. Boca (Urbana-Champaign)
MSC:
| 46L35 | Classifications of \(C^*\)-algebras |
| 19K14 | \(K_0\) as an ordered group, traces |
| 19K35 | Kasparov theory (\(KK\)-theory) |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 46M18 | Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.) |
Keywords:
\(C^*\)-algebra classification; filtered ordered \(K\)-theory; extensions; graph \(C^*\)-algebrasCitations:
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