James Gabe

A note on nonunital absorbing extensions

Elliott and Kucerovsky stated that a nonunital extension of separable C -algebras with a stable ideal is nuclearly absorbing if and only if the extension is purely large. However, their proof was flawed. We give a counterexample to their theorem as stated, but establish an equivalent formulation of nuclear absorption under a very mild additional assumption to being purely large. In particular, if the quotient algebra is nonunital, then we show that the original theorem applies. We also examine how this affects results in classification theory.

GRAPH C ∗ -ALGEBRAS WITH A T1 PRIMITIVE IDEAL SPACE

We give necessary and sufficient conditions which a graph should satisfy in order for its associated C ∗-algebra to have a T 1 primitive ideal space. We give a description of which one-point sets in such a primitive ideal space are open, and use this to prove that anypurely infinite graph C ∗-algebra purely infinite graph C ∗-algebra purely infinite graph C ∗-algebra with a T 1 (in particular Hausdorff) primitive ideal space, is a c 0-direct sum of Kirchberg algebras. Moreover, we show that graph C ∗-algebras with a T 1 primitive ideal space canonically may be given the structure of a \(C(\tilde{\mathbb{N}})\)-algebra, and that isomorphisms of their \(\tilde{\mathbb{N}}\)-filtered K-theory (without coefficients) lift to \(E(\tilde{\mathbb{N}})\)-equivalences, as defined by Dadarlat and Meyer.