On higher real and stable ranks for CCR \(C^{*}\)-algebras (Q2796098)

In this paper, arising from [\textit{L. G. Brown} and \textit{G. K. Pedersen}, J. Oper. Theory 61, No. 2, 381--417 (2009; Zbl 1212.46073)], the author obtains several results, generalizing some of \textit{V. Nistor} [J. Oper. Theory 17, 365--373 (1987; Zbl 0647.46056)], on estimating Rieffel topological stable rank and Brown-Pedersen real rank for some extensions or composition series of \(C^*\)-algebras such as CCR \(C^*\)-algebras as extensions or limits of composition series, and \(C^*\)-algebras having closed ideals of only finite-dimensional irreducible representations or of only infinite-dimensional irreducible representations, in terms of ideals and quotients, or of subquotients.NEWLINENEWLINEAlso, it is proved as a theorem generalizing a result of \textit{A. Green} [A letter (1976)] that, for a separable CCR \(C^*\)-algebra \(A\), it has generalized continuous trace (GCT) of \textit{J. Dixmier} \& Cie. (1964; Zbl 0152.32902)] if and only if the primitive ideal space of \(A\) has several equivalent conditions such as being metacompact, being a countable union of closed compact sets, and having properties with respect to closed sets and open sets, if and only if \(A\) is stably isomorphic to a \(C^*\)-algebra of only finite-dimensional irreducible representations, where the equivalence between the first and the last is due to Green.NEWLINENEWLINENote that it seems that \(K_0(A)\) is not a ring in general, but \(KK(A, A)\) is a ring via the Kasparov product.