Non-simple purely infinite \(C^{*}\)-algebras: The Hausdorff case. (English) Zbl 1048.46049
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.
Reviewer: V. M. Manuilov (Moskva)
MSC:
| 46L35 | Classifications of \(C^*\)-algebras |
| 19K99 | \(K\)-theory and operator algebras |
| 46L05 | General theory of \(C^*\)-algebras |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
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