Classification of direct limits of even Cuntz-circle algebras

TL;DR

This paper establishes a classification theorem for a broad class of purely infinite simple C*-algebras, showing that their isomorphism types are determined by K-theoretic data, including tensor products with even Cuntz algebras.

Contribution

It introduces a new classification framework for direct limits of certain C*-algebras, including tensor products with even Cuntz algebras, based on K-theory invariants.

Findings

Two unital C*-algebras in the class are isomorphic iff their K-theoretic data match.

The class includes all possible K-groups with countable odd torsion groups.

The class is closed under natural algebraic operations.

Abstract

We prove a classification theorem for purely infinte simple C*-algebras that is strong enough to show that the tensor products of two different irrational rotation algebras with the same even Cuntz algebra are isomorphic. In more detail, let C be be the class of simple C*-algebras A which are direct limits A = lim A_k, in which each A_k is a finite direct sum of algebras of the form C(X) \otimes M_n \otimes O_m, where m is even, O_m is the Cuntz algebra, X is either a point, a compact interval, or the circle S^1, and each map A_k ---> A is approximately absorbing. ("Approximately absorbing" is defined in Section 1 of the paper.) We show that two unital C*-algebras A and B in the class C are isomorphic if and only if (K_0 (A), [1_A], K_1 (A)) is isomorphic to (K_0 (B), [1_B], K_1 (B)). This class is large enough to exhaust all possible K-groups: if G_0 and G_1 are countable odd torsion groups and g is in G_0, then there is a C*-algebra A in C with (K_0 (A), [1_A], K_1 (A)) isomorphic to (G_0, g, G_1). The class C contains the tensor products of irrational rotation algebras with even Cuntz algebras. It is also closed under several natural operations.