Reciprocal Cuntz--Krieger algebras
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915172214374400 |
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| author | Matsumoto, Kengo Sogabe, Taro |
| author_facet | Matsumoto, Kengo Sogabe, Taro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18126 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reciprocal Cuntz--Krieger algebras Matsumoto, Kengo Sogabe, Taro Operator 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 $C^*$-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. |
| title | Reciprocal Cuntz--Krieger algebras |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2502.18126 |