The ranges of \(K\)-theoretic invariants for nonsimple graph algebras
The authors obtain the following main results, among many others (cf. [\textit{S. Eilers} and \textit{M. Tomforde}, Math. Ann. 346, No. 2, 393--418 (2010; Zbl 1209.46035)], [\textit{S. Eilers} et al., J. Funct. Anal. 265, No. 3, 449--468 (2013; Zbl 1294.46052); corrigendum ibid. 270, No. 2, 854--859 (2016; Zbl 1355.46051)], [\textit{S. Eilers} and \textit{G. Restorff}, in: Operator algebras. The Abel symposium 2004. Proceedings of the first Abel symposium, Oslo, Norway, September 3--5, 2004. Berlin: Springer. 87--96 (2006; Zbl 1118.46053)] and more).NEWLINENEWLINETheorem 6.1. For the class of graph \(C^*\)-algebras, each of which is an extension of an AF \(C^*\)-algebra that is the largest non-trivial ideal by a purely infinite \(C^*\)-algebra, the six-term exact sequence of \(K\)-theory groups associated to such an extension, with pre-ordered abelian groups, is a complete stable isomorphism invariant, and the range of this invariant consists of all six-term exact sequences of countable abelian groups at six positions from the trivial group to a free abelian group, to another free abelian group, to a Riesz group, to a group with trivial pre-ordering, to another group with trivial pre-ordering, and back to zero. Similarly,NEWLINENEWLINETheorem 6.2. In Theorem 6.1, replacing the largest ideal with the smallest, a similar assertion holds with the range of six-term diagrams from zero to group with the trivial per-ordering, to another with trivial or non-trivial, to a Riesz group, to a free abelian group, to another free abelian group, and back to zero.NEWLINENEWLINETheorem 6.3. In Theorem 6.1, assuming that a graph \(C^*\)-algebra has a unique non-trivial ideal, a similar assertion holds with the range of six-term diagrams in the four cases where the ideal and the quotient are either both Kirchberg \(C^*\)-algebras, or an AF \(C^*\)-algebra and a Kirchberg algebra, or a Kirchberg algebra and an AF, or both AF, respectively. Moreover,NEWLINENEWLINETheorem 6.4. For the class of graph \(C^*\)-algebras with vertices finite, containing a unique nontrivial ideal, the six-term exact sequence of \(K\)-theory groups associated to corresponding extensions, with pre-ordered abelian groups, and with order structure by the \(K\)-theory class of the unit of a graph \(C^*\)-algebra is a complete stable isomorphism invariant, and the range of this invariant consists of all six-term exact sequences of countable abelian groups with several conditions more than as mentioned above.NEWLINENEWLINETheorem 6.5. For the class of \(C^*\)-algebras of finite graphs with no sinks and with a unique nontrivial ideal, a similar asertion holds as in Theorem 6.4. What's more,NEWLINENEWLINETheorem 7.1. For a stenotic extension of two graph \(C^*\)-algebras both stable and either simple or AF, the following are equivalent: (i) The extension is a graph \(C^*\)-algebra; (ii) the extension is a graph \(C^*\)-algebra of real rank zero; (iii) two \(K\)-theory conditions hold with respect to boundary maps and positivity.NEWLINENEWLINETheorem 7.2. For a unital essential ideal-stabilized extension of two graph \(C^*\)-algebras both unital, simple, and purely infinite, the extension is a graph \(C^*\)-algebra if and only if it has real rank zero.
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