Classifying \(C^*\)-algebras via ordered, mod-\(p\) \(K\)-theory
We introduce an order structure on \(K_0(-)\oplus K_0(-; \mathbb{Z}/p)\). This group may also be thought of as \(K_0(-; \mathbb{Z}\oplus \mathbb{Z}/p)\). We exhibit new examples of real-rank zero \(C^*\)-algebras that are inductive limits of finite-dimensional and dimension-drop algebras, have the same ordered, graded \(K\)-theory with order unit and yet are not isomorphic. In fact they are not even stably shape equivalent. The order structure on \(K_0(-; \mathbb{Z}\oplus\mathbb{Z}/p)\) naturally distinguishes these algebras. The same invariant is used to give an isomorphism theorem for such real-rank zero inductive limits. As a corollary we obtain an isomorphism theorem for all real-rank zero approximately homogeneous \(C^*\)-algebras that arise from systems of bounded dimension growth and torsion-free \(K_0\) group.
real-rank zero approximately homogeneous \(C^*\)-algebras
scientific article; zbMATH DE number 931998
scientific article