Cuntz-Krieger algebras of infinite graphs and matrices

We show that the Cuntz-Krieger algebras of infinite graphs and infinite { 0 , 1 } \{0,1\} -matrices can be approximated by those of finite graphs. We then use these approximations to deduce the main uniqueness theorems for Cuntz-Krieger algebras and to compute their K K -theory. Since the finite approximating graphs have sinks, we have to calculate the K K -theory of Cuntz-Krieger algebras of graphs with sinks, and the direct methods we use to do this should be of independent interest.