Unital aligned shift equivalence and the graded classification conjecture for Leavitt path algebras
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909316110352384 |
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| author | Brix, Kevin Aguyar Dor-On, Adam Hazrat, Roozbeh Ruiz, Efren |
| author_facet | Brix, Kevin Aguyar Dor-On, Adam Hazrat, Roozbeh Ruiz, Efren |
| contents | We prove that a unital shift equivalence induces a graded isomorphism of Leavitt path algebras when the shift equivalence satisfies an alignment condition. This yields another step towards confirming the Graded Classification Conjecture. Our proof uses the bridging bimodule developed by Abrams, the fourth-named author and Tomforde, as well as a general lifting result for graded rings that we establish here. This general result also allows us to provide simplified proofs of two important recent results: one independently proven by Arnone and Va{\v s} through other means that the graded $K$-theory functor is full, and the other proven by Arnone and Cortiñas that there is no unital graded homomorphism between a Leavitt algebra and the path algebra of a Cuntz splice. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_03950 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unital aligned shift equivalence and the graded classification conjecture for Leavitt path algebras Brix, Kevin Aguyar Dor-On, Adam Hazrat, Roozbeh Ruiz, Efren Rings and Algebras Operator Algebras Primary: 16S88, 16W50, 19A49. Secondary: 37B10, 46L35 We prove that a unital shift equivalence induces a graded isomorphism of Leavitt path algebras when the shift equivalence satisfies an alignment condition. This yields another step towards confirming the Graded Classification Conjecture. Our proof uses the bridging bimodule developed by Abrams, the fourth-named author and Tomforde, as well as a general lifting result for graded rings that we establish here. This general result also allows us to provide simplified proofs of two important recent results: one independently proven by Arnone and Va{\v s} through other means that the graded $K$-theory functor is full, and the other proven by Arnone and Cortiñas that there is no unital graded homomorphism between a Leavitt algebra and the path algebra of a Cuntz splice. |
| title | Unital aligned shift equivalence and the graded classification conjecture for Leavitt path algebras |
| topic | Rings and Algebras Operator Algebras Primary: 16S88, 16W50, 19A49. Secondary: 37B10, 46L35 |
| url | https://arxiv.org/abs/2409.03950 |