The Weak Ideal Property and Topological Dimension Zero

Following up on previous work, we prove a number of results for ${{\text{C}}^{*}}$ -algebras with the weak ideal property or topological dimension zero, and some results for ${{\text{C}}^{*}}$ -algebras with related properties. Some of the more important results include the following: • The weak ideal property implies topological dimension zero. • For a separable ${{\text{C}}^{*}}$ -algebra $A$ , topological dimension zero is equivalent to $\text{RR}\left( {{\mathcal{O}}_{2}}\otimes A \right)=0$ , to $D\,\otimes \,A$ having the ideal property for some (or any) Kirchberg algebra $D$ , and to $A$ being residually hereditarily in the class of all ${{\text{C}}^{*}}$ -algebras $B$ such that ${{\mathcal{O}}_{\infty }}\otimes B$ contains a nonzero projection. • Extending the known result for ${{\mathbb{Z}}_{2}}$ , the classes of ${{\text{C}}^{*}}$ -algebras with residual $\left( \text{SP} \right)$ , which are residually hereditarily (properly) infinite, or which are purely infinite and have the ideal property, are closed under crossed products by arbitrary actions of abelian 2-groups. • If $A$ and $B$ are separable, one of them is exact, $A$ has the ideal property, and $B$ has the weak ideal property, then $A\,{{\otimes }_{\min }}\,B$ has the weak ideal property. • If $X$ is a totally disconnected locally compact Hausdorff space and $A$ is a ${{C}_{0}}\left( X \right)$ -algebra all of whose fibers have one of the weak ideal property, topological dimension zero, residual $\left( \text{SP} \right)$ , or the combination of pure infiniteness and the ideal property, then $A$ also has the corresponding property (for topological dimension zero, provided $A$ is separable). • Topological dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable ${{\text{C}}^{*}}$ -algebras, including all separable locally $\text{AH}$ algebras. • The weak ideal property does not imply the ideal property for separable $Z$ -stable ${{\text{C}}^{*}}$ -algebras. We give other related results, as well as counterexamples to several other statements one might conjecture.