Graph $C^\ast$-algebras with a $T_1$ primitive ideal space

TL;DR

This paper characterizes when graph $C^*$-algebras have a $T_1$ primitive ideal space, describing their structure and showing how their $K$-theory invariants relate to $E$-theory equivalences.

Contribution

It provides necessary and sufficient conditions for $T_1$ primitive ideal spaces in graph $C^*$-algebras and links their structure to $K$-theory and $E$-theory.

Findings

Graph $C^*$-algebras with $T_1$ primitive ideal space are $c_0$-direct sums of Kirchberg algebras.

Such algebras can be given a $C( ilde{N})$-algebra structure.

Isomorphisms of their filtered $K$-theory lift to $E( ilde{N})$-equivalences.

Abstract

We give necessary and sufficient conditions which a graph should satisfy in order for its associated $C^\ast$-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 any purely infinite graph $C^\ast$-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 $C^\ast$-algebras with a $T_1$ primitive ideal space canonically may be given the structure of a $C(\widetilde{\mathbb N})$-algebra, and that isomorphisms of their $\widetilde{\mathbb N}$-filtered $K$-theory (without coefficients) lift to $E(\widetilde{\mathbb N})$-equivalences, as defined by Dadarlat and Meyer.