Article contents
The Weak Ideal Property and Topological Dimension Zero
Published online by Cambridge University Press: 20 November 2018
Abstract
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.
Keywords
- Type
- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 2017
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190725024643405-0884:S0008414X00015273:S0008414X00015273_inline4.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190725024643405-0884:S0008414X00015273:S0008414X00015273_inline5.gif?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190725024643405-0884:S0008414X00015273:S0008414X00015273_inline6.gif?pub-status=live)
- 6
- Cited by