Non-simple purely infinite \(C^{*}\)-algebras: The Hausdorff case. (Q1427635)

Local and global definitions of weak/strong 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 finite-dimensional, if \(A\) has real rank zero or if \(A\) is approximately divisible. Sufficient criteria are given for local pure infiniteness of tensor products. It is also shown that \(A\) is isomorphic to \(A\otimes{\mathcal O}_\infty\) if \(A\) is purely infinite separable stable nuclear and if Prim\((A)\) is Hausdorff.