# The weak ideal property and topological dimension zero

Cornel Pasnicu, N. Christopher Phillips

Canad. J. Math., Vol. 69 (6), (2017), 1385-1421

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.

###### Cornel Pasnicu

Professor

cornel.pasnicu@utsa.edu
(210) 458-4495

FLN 4.01.32