Homotopy of product systems and K-theory of Cuntz-Nica-Pimsner algebras

Abstract

We introduce the notion of a homotopy of product systems, and show that the Cuntz-Nica-Pimsner algebras of homotopic product systems over N k have isomorphic K-theory. As an application, we give a new proof that the K-theory of a 2-graph C *algebra is independent of the factorisation rules, and we further show that the K-theory of any twisted k-graph C * -algebra is independent of the twisting 2-cocycle. We also explore applications to K-theory for the C * -algebras of single-vertex k-graphs, reducing the question of whether the K-theory is independent of the factorisation rules to a question about path-connectedness of the space of solutions to an equation of Yang-Baxter type.

Remarks On the K-theory of C*-Algebras of Products of Odometers

“…At the end of [2] Barlak, Omland, and Stammeier made a conjecture to the torsion part which is equivalent to a conjecture of k-graph C*-algebras about whether the K-theory of the single-vertex k-graph C*-algebra is independent of the factorization rule. Very recently, many authors also found the kgraph algebra conjecture is highly connected with the famous Yang-Baxter equation (see for example [8,26]). Therefore, the conjecture of Barlak, Omland, and Stammeier and the conjecture about the k-graph C*-algebra are extremely important in many ways.…”