Reciprocal Cuntz--Krieger algebras

TL;DR

This paper constructs a dual algebra for Kirchberg algebras using K-theoretic duality, providing explicit realizations for Cuntz--Krieger algebras and exploring their automorphism groups and gauge actions.

Contribution

It introduces a concrete construction of the reciprocal algebra for Kirchberg algebras with finitely generated K-groups, especially for Cuntz--Krieger algebras, and analyzes their automorphisms.

Findings

Reciprocal algebra $\uhat{\uA}$ is realized as a universal $C^*$-algebra with specific generators.

An isomorphism between the fundamental groups of automorphism groups is established.

Gauge actions on reciprocal algebras are studied and related to automorphism groups.

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.