SIMPLICITY OF CUNTZ–KRIEGER ALGEBRAS OF INFINITE MATRICES

We give necessary and sufficient conditions for simplicity of Cuntz-Krieger algebras corresponding to infinite 0–1 ma-trices and of C∗-algebras corresponding to countable directed graphs. We show that simple algebras within these two classes are either purely infinite or AF. 0. Since their invention about twenty years ago Cuntz-Krieger algebras OA corresponding to finite, square 0–1 matrices [3] have attracted immense interest. It was remarked in their original paper by Cuntz and Krieger that the theory may be extended to infinite matrices as well. Unfortunately, no details were provided at that time. This quite non-trivial task has been successfully carried out recently by Exel and Laca [5], by means of some heavy-duty machinery. It turns out that the resulting class of C∗-algebras is rich enough to encompass, at least up to Morita equivalence, all graph C∗-algebras [10, 9, 1, 11], as shown in [6], as well as AF-algebras, as shown in [9, 4]. Despite a lot of interest in the subject and a number of well-aime