The C∗-algebras of infinite graphs

Abstract. We associate C-algebras to innite directed graphs that are not necessarily locally nite. 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 nite directed graphs. A directed graph E = (E0; E1; r; s) consists of a set E0 of vertices, a set E1 of edges, and range and source maps r; s: E1! E0. The C-algebra of E is the universal C-algebra C(E) generated by mutually orthogonal projections fpv: v 2 E0g and partial isometries fse: e 2 E1g which satisfy sese = pr(e) and pv = X e:s(e)=v ses e:(1.1) For nite graphs, these are precisely the Cuntz-Krieger algebras OA: given E, tak