Cuntz–Krieger algebras of directed graphs

Abstract

We associate to each row-finite directed graph E a universal Cuntz-Krieger C * -algebra C * (E), and study how the distribution of loops in E affects the structure of C * (E). We prove that C * (E) is AF if and only if E has no loops. We describe an exit condition (L) on loops in E which allows us to prove an analogue of the Cuntz-Krieger uniqueness theorem and give a characterisation of when C * (E) is purely infinite. If the graph E satisfies (L) and is cofinal, then we have a dichotomy: if E has no loops, then C * (E) is AF; if E has a loop, then C * (E) is purely infinite. , The ideal structure of groupoid crossed product C * -algebras, J. Operator Theory, 25 (1991), 3-36.

On graphC*-algebras

“…This situation has been studied by Kumjian, Pask and Raeburn (see [14,Corollary 3.10.]). Using a groupoid approach, they carry out a detailed analysis of how the distribution of loops affects the structure of 6 V , for any row-finite graph F. [3] On graph C*-algebras 155…”

Complex Kumjian-Pask algebras of 2-graphs

“…Higher-rank graphs ork-graphs were firstly introduced by Kumjian and Pask in [1]. They are higher-dimensional analogues of directed graphs.…”

“…Note that, is a semigroup under addition with identity which can be regarded as the morphisms in a category with one object and composition given by addition, we denote this category as .We recall from( [1], Definition 1) a -graph (higher-rank graph or rank graph) is a countable category together with a functor (called the degree map) which satisfies the factorization property: for every morphism and every decomposition with , there exist unique morphism and such that , , and…”

“…They generalized the result of [4] to higher rank graph by describingKumjian-Pask relation algebraic analog of Cuntz-Krieger relation of [1] and constructingKumjian-Pask algebra as a quotient of the free -algebra on the set of paths in .In the following years,Rosjanuardi in [10] has investigated the relationship between Kumjian-Pask algebra (particularly, for and got analogue result of [6] for higher rank graph. Generally, the results on KumjianPask algebra were initiated by the works on Leavitt-path algebras of [4].…”

Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

“…We recall some terminology for graphs (see, for example, [12,2]; note that our edge direction convention agrees with that used in these papers and in [11], which is opposite to that used in [14]). …”