Dr. Pasnicu was invited to present at the 2018 Symposium on K-Theory and Non-Commutative Topology in San Juan, Puerto Rico.
In his presentation he discussed The Weak Ideal Property and Topological Dimension Zero.
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 resu
•The weak ideal property implies topological dimension zero.
• For a separable C*-algebra A, topological dimension zero is equivalent to RR(O2 ⌦ A) = 0, to D⌦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• ⌦Bcontains a nonzero projection.
• Extending the known result for Z2, the classes of C*-algebras with residual (SP), 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⌦min B has the weak ideal property.
• If X is a totally disconnected locally compact Hausdorff space and A is a C0(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.
This is joint work with N. Christopher Phillips and it appeared in the Canad. J. Math.
For More Information on the Symposium click the link below