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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.19067 |
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| _version_ | 1866916338102960128 |
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| author | Ruiz, Efren |
| author_facet | Ruiz, Efren |
| contents | The Algebraic Kirchberg-Phillips Question for Leavitt path algebras asks whether unital $K$-theory is a complete isomorphism invariant for unital, simple, purely infinite Leavitt path algebras over finite graphs. Most work on this problem has focused on determining whether (up to isomorphism) there is a unique unital, simple, Leavitt path algebra with trivial $K$-theory (often reformulated as the question of whether the Leavitt path algebras $L_2$ and $L_{2_-}$ are isomorphic). However, it is unknown whether a positive answer to this special case implies a positive answer to the Algebraic Kirchberg-Phillips Question. In this note, we pose a different question that asks whether two particular non-simple Leavitt path algebras $L_k(\mathbf{F}_*)$ and $L_k(\mathbf{F}_{**})$ are isomorphic, and we prove that a positive answer to this question implies a positive answer to the Algebraic Kirchberg-Phillips Question. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19067 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Algebraic Kirchberg-Phillips Question for Leavitt path algebras Ruiz, Efren Rings and Algebras Primary: 16S88, Secondary: 46L35, 37B10 The Algebraic Kirchberg-Phillips Question for Leavitt path algebras asks whether unital $K$-theory is a complete isomorphism invariant for unital, simple, purely infinite Leavitt path algebras over finite graphs. Most work on this problem has focused on determining whether (up to isomorphism) there is a unique unital, simple, Leavitt path algebra with trivial $K$-theory (often reformulated as the question of whether the Leavitt path algebras $L_2$ and $L_{2_-}$ are isomorphic). However, it is unknown whether a positive answer to this special case implies a positive answer to the Algebraic Kirchberg-Phillips Question. In this note, we pose a different question that asks whether two particular non-simple Leavitt path algebras $L_k(\mathbf{F}_*)$ and $L_k(\mathbf{F}_{**})$ are isomorphic, and we prove that a positive answer to this question implies a positive answer to the Algebraic Kirchberg-Phillips Question. |
| title | The Algebraic Kirchberg-Phillips Question for Leavitt path algebras |
| topic | Rings and Algebras Primary: 16S88, Secondary: 46L35, 37B10 |
| url | https://arxiv.org/abs/2407.19067 |