On universal property of reciprocal Kirchberg algebras and uniquely ergodic automorphisms

TL;DR

This work establishes a K-theoretic duality framework for unital Kirchberg algebras with finitely generated $K$-groups by constructing the reciprocal dual ${\widehat{\mathcal{A}}}$ as a corner of a Cuntz–Pimsner algebra and proving a universal presentation in terms of a generating subalgebra ${\mathcal{T}}$ and a family of partial isometries. It proves a parallel universality for ${\widehat{\mathcal{A}}}$ and derives an explicit aperiodic ergodic automorphism on such algebras, which has a unique invariant state that is pure. The theory is illustrated with concrete Exel–Laca realizations for reciprocal duals of simple Cuntz–Krieger algebras and related algebras, including a canonical ergodic automorphism on ${\mathcal{O}}_2$ and its unique invariant state. These results deepen the connection between K-theoretic duality and dynamical properties of Kirchberg algebras, with potential implications for automorphism groups and classification. The approach leverages universal properties of ${\mathcal{O}}_{H_{\mathcal{F}}}$ and corner techniques to translate abstract dualities into concrete generator-relator descriptions.

Abstract

Reciprocality in Kirchberg algebras with finitely generated K-groups is regarded as a K-theoretic duality through K-groups and strong extension groups. We will prove that the reciprocal Kirchberg algebra has a universal property with respect to some generating C*-subalgebra and a family of generating partial isometries. By using the universal property, we will prove that there exists an aperiodic ergodic automorphism on an arbitrary unital Kirchberg algebra with finitely generated K-groups, which has a unique invariant state. The state is pure.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 1.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof