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Reciprocal Cuntz--Krieger algebras

Kengo Matsumoto, Taro Sogabe

TL;DR

The paper develops a systematic construction of the reciprocal dual ${\widehat{\mathcal{A}}}$ for Kirchberg algebras ${\mathcal{A}}$ with finitely generated $ ext{K}$-groups, tying Ext-theoretic and $ ext{K}$-theoretic data through strong duality for extensions. It provides concrete realizations of the reciprocal dual for simple Cuntz–Krieger algebras ${\mathcal{O}}_A$ by exhibiting ${\widehat{\mathcal{O}}_A}$ as a corner of a Toeplitz/Exel–Laca framework or as a corner of an Exel–Laca algebra ${\mathcal{O}}_{\widehat{A}_\infty}$, with explicit generators, relations, and universal descriptions. The work analyzes gauge actions on ${\mathcal{O}}_{A^\infty}$ and ${\widehat{\mathcal{O}}_A}$, showing a unique ground state for the $\gamma$-action and absence of KMS states, and proves a fundamental isomorphism between the first fundamental groups of automorphism groups: $\pi_1(\mathrm{Aut}(\mathcal{O}_A)) \cong \pi_1(\mathrm{Aut}(\widehat{\mathcal{O}}_A))$ sending the standard gauge class to its reciprocal. These results position reciprocal duality as a concrete invariant-enhancing tool for classifying Kirchberg algebras with finitely generated $ ext{K}$-groups and illuminate the gauge-structure correspondence across duals.

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 $π_1({\operatorname{Aut}}({\mathcal{O}}_A))$ and $π_1({\operatorname{Aut}}(\widehat{\mathcal{O}}_A))$ preserving their gauge actions.

Reciprocal Cuntz--Krieger algebras

TL;DR

The paper develops a systematic construction of the reciprocal dual for Kirchberg algebras with finitely generated -groups, tying Ext-theoretic and -theoretic data through strong duality for extensions. It provides concrete realizations of the reciprocal dual for simple Cuntz–Krieger algebras by exhibiting as a corner of a Toeplitz/Exel–Laca framework or as a corner of an Exel–Laca algebra , with explicit generators, relations, and universal descriptions. The work analyzes gauge actions on and , showing a unique ground state for the -action and absence of KMS states, and proves a fundamental isomorphism between the first fundamental groups of automorphism groups: sending the standard gauge class to its reciprocal. These results position reciprocal duality as a concrete invariant-enhancing tool for classifying Kirchberg algebras with finitely generated -groups and illuminate the gauge-structure correspondence across duals.

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 for a Kirchberg algebra with finitely generated K-groups via K-theoretic duality for extensions. In particular, we may concretely realize the reciprocal algebra for simple Cuntz--Krieger algebras . As a result, the algebra is realized as a unital simple purely infinite universal -algebra generated by a family of partial isometries subject to certain operator relations. We will also study gauge actions on the reciprocal algebra and prove that there exists an isomorphism between the fundamental groups and preserving their gauge actions.

Paper Structure

This paper contains 23 sections, 40 theorems, 185 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Introduction
  3. Preliminaries
  4. Reciprocal duality and $\operatorname{K}$-theoretic duality for extensions
  5. Reciprocal duality
  6. $\operatorname{K}$-theoretic duality for extensions
  7. General construction of the reciprocal dual
  8. The reciprocal dual of Cuntz--Krieger algebras
  9. The comparison between $\widehat{\mathcal{O}}_A$ and ${{\mathcal{O}}_A}$
  10. The Cuntz--Krieger--Toeplitz extensions
  11. A concrete realization of $\widehat{\mathcal{O}}_A$
  12. Realization of ${{\mathcal{O}}_{A^\infty}}$ as a Exel--Laca algebra
  13. Realization of $\mathcal{O}_{A^\infty}$ as a free product
  14. Generators and relations of $\widehat{\mathcal{O}}_A$
  15. Generators of $\widehat{\mathcal{O}}_A$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let ${\mathcal{A}}$ be a Kirchberg algebra with finitely generated $\operatorname{K}$-groups. For any essential unital extension $0 \rightarrow \mathcal{K} \rightarrow \mathcal{E} \rightarrow {\mathcal{A}} \rightarrow 0$ of ${\mathcal{A}}$ and $\epsilon \in \{-1,1\}$, we have a systematic constructi where $[\mathcal{E}]_s\in \operatorname{Ext}_{\operatorname{s}}^1({\mathcal{A}})$ is the strong equ

Theorems & Definitions (71)

  • Theorem 1.1: Theorem \ref{['thm:construction']}
  • Theorem 1.2: Theorem \ref{['thm:mr']}, Corollary \ref{['cor:whatOA']}, Proposition \ref{['prop:ExelLaca']} and Theorem \ref{['thm:freeproduct']}
  • Theorem 1.3: Theorem \ref{['thm:univrelation']}
  • Theorem 1.4: Theorem \ref{['thm:gaugeinpi1whatOA']}
  • Definition 2.1: Sogabe2022
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Lemma 2.4: MatSogabe2
  • Definition 2.5: MaJMAA2024, PennigSogabe
  • ...and 61 more