Classification of non-simple purely infinite C*-algebras
Final Report Abstract
Noncommutative topology considers C∗ -algebras as appropriate generalisations of locally compact Hausdorff spaces. Some classes of C∗ -algebras are very rigid, that is, they may be classified by rather simple K-theoretic invariants. This project has developed methods to classify non-simple, purely infinite C∗ -algebras with suitable ideal structure. The invariant that is computed in this project is applicable in many cases, imposing much weaker assumptions on the ideal structure of the C∗ -algebras. Many C∗ -algebras are defined by dynamical systems or combinatorial data of some kind. The construction of Cuntz–Krieger algebras is an example of this. They are closely related to certain dynamical systems, and their K-theoretic invariants are also often useful to classify these dynamical systems. This project proved a theorem that describes when a purely infinite C∗ -algebra is isomorphic to a Cuntz–Krieger algebra. Graph algebras generalise Cuntz–Krieger algebras and are defined by combinatorial data, namely, directed graphs. The same C∗ -algebra may be defined by many different graphs. The project clarified the relationship between graphs with isomorphic graph C∗ -algebras by showing that a certain modification of graphs – the Cuntz splice – preserves the isomorphism class of the graph C∗ -algebras if both are purely infinite.
Publications
- Extensions of Cuntz–Krieger algebras (2015)
Rasmus Bentmann
- A more general method to classify up to equivariant KK-equivalence, Doc. Math. 22 (2017), 423–454
Rasmus Bentmann and Ralf Meyer
(See online at https://doi.org/10.25537/dm.2017v22.423-454) - A more general method to classify up to equivariant KK-equivalence II: Computing obstruction classes (2018)
Ralf Meyer
- Cuntz splice invariance for purely infinite graph algebras, Math. Scand. 122 (2018), no. 1, 91–106
Rasmus Bentmann
(See online at https://doi.org/10.7146/math.scand.a-96633)
