$k$-graph algebras are iterated Cuntz-Pimsner algebras -- from the bottom up
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908981823275008 |
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| author | Deaconu, Valentin Paulovicks, Menevşe Eryüzlü Kaliszewski, S. Quigg, John |
| author_facet | Deaconu, Valentin Paulovicks, Menevşe Eryüzlü Kaliszewski, S. Quigg, John |
| contents | We introduce a new method of expressing a $k$-graph $C^*$-algebra as a Cuntz-Pimsner algebra. Kumjian, Pask, and Sims have done this directly, using a linking algebra approach and a $(k-1)$-graph algebra. This can be iterated downward. Our process, on the other hand, starts at the bottom, with Pimsner's theorem for graph algebras, and iterates upward. We actually work with product systems over $\mathbb N^k$, and the result for $k$-graphs is a special case. Our iteration step involves a ``decategorization'' of a recent theorem showing that the Cuntz-Pimsner construction is functorial at the level of ``enchilada categories''. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20923 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $k$-graph algebras are iterated Cuntz-Pimsner algebras -- from the bottom up Deaconu, Valentin Paulovicks, Menevşe Eryüzlü Kaliszewski, S. Quigg, John Operator Algebras Primary 46L05, Secondary 46M15 We introduce a new method of expressing a $k$-graph $C^*$-algebra as a Cuntz-Pimsner algebra. Kumjian, Pask, and Sims have done this directly, using a linking algebra approach and a $(k-1)$-graph algebra. This can be iterated downward. Our process, on the other hand, starts at the bottom, with Pimsner's theorem for graph algebras, and iterates upward. We actually work with product systems over $\mathbb N^k$, and the result for $k$-graphs is a special case. Our iteration step involves a ``decategorization'' of a recent theorem showing that the Cuntz-Pimsner construction is functorial at the level of ``enchilada categories''. |
| title | $k$-graph algebras are iterated Cuntz-Pimsner algebras -- from the bottom up |
| topic | Operator Algebras Primary 46L05, Secondary 46M15 |
| url | https://arxiv.org/abs/2603.20923 |