Reciprocal Cuntz--Krieger algebras

Abstract

Reciprocality in Kirchberg algebras is a duality between strong extension groups and K-theory groups. We describe a construction of the reciprocal dual algebra $\widehat{\mathcal{A}}$ for a Kirchberg algebra $\mathcal{A}$ with finitely generated K-groups via K-theoretic duality for extensions. In particular, we may concretely realize the reciprocal algebra $\widehat{\mathcal{O}}_A$ for simple Cuntz--Krieger algebras $\mathcal{O}_A$. As a result, the algebra $\widehat{\mathcal{O}}_A$ is realized as a unital simple purely infinite universal $C^*$-algebra generated by a family of partial isometries subject to certain operator relations. We will also study gauge actions on the reciprocal algebra $\widehat{\mathcal{O}}_A$ and prove that there exists an isomorphism between the fundamental groups $\pi_1({\operatorname{Aut}}({\mathcal{O}}_A))$ and $\pi_1({\operatorname{Aut}}(\widehat{\mathcal{O}}_A))$ preserving their gauge actions.