Classification of certain infinite simple \(C^*\)-algebras
The author shows that the class of \(C^*\)-algebras, that arise as the crossed product of a stable simple AF-algebra with a \(\mathbb{Z}\)-action determined by an automorphism, which maps a projection in the algebra onto a proper subprojection, consists of simple, purely infinite \(C^*\)- algebras, and he shows that a specific subclass of it is classified by \(K\)-theory. This subclass is large enough to exhaust all possible \(K\)- groups: if \(G_0\) and \(G_1\) are countable abelian groups, with \(G_1\) torsion free (as it must be), then there is a \(C^*\)-algebra \(A\) in the classified subclass with \(K_0(A)\cong G_0\) and \(K_1(A)\cong G_1\). The subclass contains the Cuntz algebras \({\mathcal O}_n\), with \(n\) even, and the Cuntz-Krieger algebras \({\mathcal O}_A\), with \(K_0({\mathcal O}_A)\) of finite odd order, and it is closed under forming inductive limits. The \(C^*\)-algebras in the classified subclass can be viewed as classifiable models (in a strong sense) of general, simple purely infinite \(C^*\)-algebras with the same \(K\)-theory.
crossed product of a stable simple AF-algebra with a \(\mathbb{Z}\)-action
simple, purely infinite \(C^*\)-algebras
scientific article; zbMATH DE number 790370