Reciprocal Cuntz--Krieger algebras

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 $C^{\ast }$-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 ${\pi }_1(\mathrm{Aut}({\mathcal{O}}_A))$ and ${\pi }_1(\mathrm{Aut}({\widehat{\mathcal{O}}}_A))$ preserving their gauge actions.