The 𝐶*-algebras of infinite graphs

Abstract

Abstract. We associate C * -algebras to infinite directed graphs that are not necessarily locally finite. By realizing these algebras as Cuntz-Krieger algebras in the sense of Exel and Laca, we are able to give criteria for their uniqueness and simplicity, generalizing results of Kumjian, Pask, Raeburn, and Renault for locally finite directed graphs.of edges, and range and source maps r, s : For finite graphs, these are precisely the Cuntz-Krieger algebras O A : given E, take A to be the edge matrix A E defined byconversely, given an I × I matrix A, form the graph E A with vertex set I and incidence matrix A, and then C * (E A ) is isomorphic to O A in a slightly nonobvious way (see [9, Theorem 3] or [13, Proposition 4.1]). This correspondence carries over to locally finite graphs (graphs in which vertices receive and emit finitely many edges), and the classical uniqueness and simplicity theorems for Cuntz-Krieger algebras have elegant extensions to these graph algebras [11,12].If a vertex v emits infinitely many edges, then the Cuntz-Krieger relation (1.1) does not make sense in an abstract C * -algebra. On the other hand, Fowler and Raeburn have recently noticed that if all vertices emit infinitely many edges, one can obtain uniqueness and simplicity theorems for graph algebras without demanding equality in (1.1) [8, Corollaries 4.2 and 4.5]. This suggests that a satisfactory theory should be possible if we merely insist that (1.1) holds when v emits finitely many edges.Exel and Laca have studied a generalization of the Cuntz-Krieger algebras which allows arbitrary infinite matrices, and have obtained uniqueness and simplicity theorems among other interesting results [7]. We shall show that, if we define C * (E) in the way suggested by [8], then we can pass from the graph algebra C * (E)

C*-algebras associated with presentations of subshifts ii. ideal structure and lambda-graph subsystems

“…-graph system £ comes from a finite directed graph G, the associated C*algebra O& becomes a Cuntz-Krieger algebra O Ac for its adjacency matrix A c with entries in {0, 1}. The results of this paper, Theorem A, Corollary B, [2,10,12,15,17,18,30,34] and [35]. Related discussions for C*-algebras generated by Hilbert C*-bimodules can be found in [14].…”

Visualizing Automorphisms of Graph Algebras

“…Graph algebras provide a natural generalization of the Cuntz-Krieger algebras [15] and are the subject of widespread investigation by specialists in the theory of operator algebras, symbolic dynamics, non-commutative geometry and quantum groups. The graph C * -algebra [16,23,24] of E, denoted C * (E), is the universal C * -algebra generated by a collection of mutually orthogonal projections {P v | v ∈ E 0 } and partial isometries {S e | e ∈ E 1 } satisfying the relations…”

Invariance of the Cuntz splice

“…Graph C * -algebras. We follow the notation and definition for graph C * -algebras in [FLR00]; this is not the convention used in Raeburn's monograph [Rae05]. Definition 2.6.…”