Graph C*-Algebras with a T₁ Primitive Ideal Space

We give necessary and sufficient conditions which a graph should satisfy in order for its associated C∗-algebra to have a T1 primitive ideal space. We give a description of which one-point sets in such a primitive ideal space are open, and use this to prove that anypurely infinite graph C∗-algebrapurely infinite graph C∗-algebra purely infinite graph C∗-algebra with a T1 (in particular Hausdorff) primitive ideal space, is a c0-direct sum of Kirchberg algebras. Moreover, we show that graph C∗-algebras with a T1 primitive ideal space canonically may be given the structure of a C(N ~ ) -algebra, and that isomorphisms of their N ~ -filtered K-theory (without coefficients) lift to E(N ~ ) -equivalences, as defined by Dadarlat and Meyer