Non-simple purely infinite C∗-algebras: the Hausdorff case

Local and global definitions of pure infiniteness for a C∗-algebra A are compared, and equivalence between them is obtained if the primitive ideal space of A is Hausdorff and of finite dimension, if A has real rank zero, or if A is approximately divisible. Sufficient criteria are given for local pure infiniteness of tensor products. They yield that exact simple tensorially non-prime C∗-algebras are purely infinite if they have no semi-finite lower semi-continuous trace. One obtains that A is isomorphic to A⊗O∞ if A is (1-)purely infinite, separable, stable, nuclear and Prim(A) is a Hausdorff space (not necessarily of finite dimension).