On graph \(C^*\)-algebras with a linear ideal lattice
The context for the work in this paper is the classification of graph \(C^*\)-algebras with finitely many ideals. The authors introduce a ``working conjecture'' saying that graph \(C^*\)-algebras \(C^*(E)\) with finitely many ideals are classified up to stable isomorphism by their filtrated, ordered \(K\)-theory \(FK_X^+(C^*(E)\). Recall that for a non-simple graph \(C^*\)-algebra \(C^*(E)\) with \(X=\text{Prim}(C^*(E))\), this invariant consists of all \(K_0\) and \(K_1\) groups of subquotients of \(C^*(E)\), including the natural transformations between them, and the order on each \(K_0\) group. Here the authors look at the case of \(C^*(E)\) with a linear ideal structure \(0=\mathfrak{I}_0\triangleleft \mathfrak{I}_1 \cdots \triangleleft \mathfrak{I}_n=C^*(E)\). The main point of the present paper is to present a situation where \(FK_X^+(C^*(E)\) is much simplified, for instance the connecting maps carry no information. More precisely, Theorem 4.1 is a stable classification result for separable, nuclear \(C^*\)-algebras both with linear ideal structures and with the following \(K\)-theoretic information: the \(K_0\)-groups of intermediate subquotients are free while the \(K_1\)-groups of the same subquotients are trivial, the intermediate subquotients are purely infinite simple \(C^*\)-algebras satisfying the UCT or are simple AF algebras, and a technical condition on fullness of extensions. Their classification result covers the class of graph algebras \(C^*(E)\) with a finite ideal lattice in the case that \(E\) has finitely many vertices and is such that two vertices which are connected by an edge are connected by infinitely many edges.
scientific article; zbMATH DE number 5712980