## Cornel Pasnicu: On the weak ideal property and some related properties, 147-157

### Abstract:

Let be any of the following properties: the weak ideal property, the topological dimension zero, the combination of pure infiniteness and the ideal property, residual (SP), pure infiniteness, strong pure infiniteness, provided that the zero C*-algebra is included, and -stability for a separable unital strongly self-absorbing C*-algebra . Let be a separable unital C*-algebra with unit , and assume that there exist non-zero projections , in such that . We show that has     has for every      has  (in fact, we prove a much more general result). We also show that, somehow surprisingly, for two large classes of non-zero C*-algebras, if , then the fact that has the weak ideal property implies (or, is equivalent to) the fact that and have the weak ideal property. We prove that one of these two results still holds if we replace the weak ideal property by some related properties.

Key Words: Weak ideal property, topological dimension zero, ideal property, tensor product C*-algebra, type I C*-algebra.

2010 Mathematics Subject Classification: Primary 46L06; Secondary 46L05.