Graph C*-Algebras with Real Rank Zero

Real rank of C*-algebras associated with graphs

“…• Recall that for a subgraph F of E the loop completion t E {F) is the subgraph of E obtained by adding all the loops based at vertices of F° to the graph F, and the loop contraction tc{F) of F is the graph obtained by shrinking each loop in F to a loop consisting of a single edge, [8 (2) => (1). By considering the desingularization we may assume that £ is a rowfinite graph with no sinks.…”

“…Recently, for graph C*-algebras their ideal structures, simplicity criteria, and their K-theory have been studied by many authors (see [1,2,6,[8][9][10][11]13] among others), and we know from [9] and [8] that if £ is a locally finite directed graph then its graph C*-algebra C*(E) has real rank zero exactly when the graph E satisfies loop condition (K), which implies that for a Cuntz-Krieger algebra 6 B , where B is the edge matrix of a finite graph E, RR{0 B ) = 0 if and only if the matrix B satisfies condition (II) considered in [4]. The purpose of this paper is to generalize this result to an arbitrary graph E (see Theorem 3.5).…”

“…In view of the fact that the C'-algebras C*(E) associated with arbitrary graphs are much harder to understand than those associated with locally finite ones it would be useful to present the proof of Theorem 3.5 here applying several recently known generalized facts for arbitrary graphs (or row-finite graphs) to handle the general case even though the main idea of the proof is basically same as in [8,9]. [3]…”

Real rank and topological dimension of higher rank graph algebras

“…A C * -algebra A has real-rank zero if the set of invertible self-adjoint elements of the minimal unitisation of A is dense in the set of all self-adjoint elements in the minimal unitisation of A [4]. For k = 1, it was established in the locally finite case in [14,15], and in general by Hong and Szymański in [13,Theorem 2.5], that a graph C * -algebra C * (E) has real-rank zero if and only if the graph satisfies Condition (K). However, the methods of [14,15,13] do not generalise to k-graphs.…”

“…For k = 1, it was established in the locally finite case in [14,15], and in general by Hong and Szymański in [13,Theorem 2.5], that a graph C * -algebra C * (E) has real-rank zero if and only if the graph satisfies Condition (K). However, the methods of [14,15,13] do not generalise to k-graphs. So we restrict our attention to the special case of k-graphs Λ for which C * (Λ) is purely infinite.…”