The weak ideal property and topological dimension zero

TL;DR

This paper establishes deep connections between the weak ideal property and topological dimension zero in C*-algebras, proving implications, equivalences, and stability under various constructions, with counterexamples clarifying limitations.

Contribution

It proves that the weak ideal property implies topological dimension zero and characterizes topological dimension zero in terms of other properties, extending known results and providing new stability and counterexample results.

Findings

Weak ideal property implies topological dimension zero.

Topological dimension zero is equivalent to several other properties for separable C*-algebras.

Certain properties are preserved under crossed products and tensor products with specific conditions.

Abstract

Following up on previous work, we prove a number of results for C*-algebras with the weak ideal property or topological dimension zero, and some results for C*-algebras with related properties. Some of the more important results include: The weak ideal property implies topological dimension zero. For a separable C*-algebra~A, topological dimension zero is equivalent to RR (O_2 \otimes A) = 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 C*-algebras B such that O_{\infty} \otimes B contains a nonzero projection. Extending the known result for Z_2, the classes of C*-algebras with topological dimension zero, with the weak ideal property, and with residual (SP) 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 (X)-algebra all of whose fibers have one of the weak ideal property, topological dimension zero, residual (SP), 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 C*-algebras including all separable locally AH algebras. The weak ideal property does not imply the ideal property for separable Z-stable C*-algebras. We give other related results, as well as counterexamples to several other statements one might hope for.