Total Cuntz semigroup, Extension and Elliott Conjecture with Real rank zero
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929476821057536 |
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| author | An, Qingnan Liu, Zhichao |
| author_facet | An, Qingnan Liu, Zhichao |
| contents | In this paper, we exhibit two unital, separable, nuclear ${\rm C}^*$-algebras of stable rank one and real rank zero with the same ordered scaled total K-theory, but they are not isomorphic with each other, which forms a counterexample to Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of total K-theory. For the general setting, with a new invariant -- total Cuntz semigroup \cite{AL}, we classify a large class of ${\rm C}^*$-algebras obtained from extensions. The total Cuntz semigroup, which distinguish the algebras of our counterexample, could possibly classify all the ${\rm C}^*$-algebras of stable rank one and real rank zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_06835 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Total Cuntz semigroup, Extension and Elliott Conjecture with Real rank zero An, Qingnan Liu, Zhichao Operator Algebras In this paper, we exhibit two unital, separable, nuclear ${\rm C}^*$-algebras of stable rank one and real rank zero with the same ordered scaled total K-theory, but they are not isomorphic with each other, which forms a counterexample to Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of total K-theory. For the general setting, with a new invariant -- total Cuntz semigroup \cite{AL}, we classify a large class of ${\rm C}^*$-algebras obtained from extensions. The total Cuntz semigroup, which distinguish the algebras of our counterexample, could possibly classify all the ${\rm C}^*$-algebras of stable rank one and real rank zero. |
| title | Total Cuntz semigroup, Extension and Elliott Conjecture with Real rank zero |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2303.06835 |