Bijective rigidity of uniform Roe algebras and injectivity of the comparison map
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910061096337408 |
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| author | Krutoy, Kostyantyn |
| author_facet | Krutoy, Kostyantyn |
| contents | We show that, for uniformly locally finite metric spaces $X$ and $Y$ with isomorphic uniform Roe algebras $C^*_u(X)$ and $C^*_u(Y)$, the existence of a bijective coarse equivalence $f \colon X \to Y$ is equivalent to the injectivity of the $0$th comparison map appearing in the HK conjecture for coarse groupoids. We further prove that the $0$th comparison map is injective unconditionally. Moreover, if the underlying space is coarsely connected, this map is in fact split-injective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06720 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bijective rigidity of uniform Roe algebras and injectivity of the comparison map Krutoy, Kostyantyn Operator Algebras Metric Geometry 46L05, 51F30 We show that, for uniformly locally finite metric spaces $X$ and $Y$ with isomorphic uniform Roe algebras $C^*_u(X)$ and $C^*_u(Y)$, the existence of a bijective coarse equivalence $f \colon X \to Y$ is equivalent to the injectivity of the $0$th comparison map appearing in the HK conjecture for coarse groupoids. We further prove that the $0$th comparison map is injective unconditionally. Moreover, if the underlying space is coarsely connected, this map is in fact split-injective. |
| title | Bijective rigidity of uniform Roe algebras and injectivity of the comparison map |
| topic | Operator Algebras Metric Geometry 46L05, 51F30 |
| url | https://arxiv.org/abs/2602.06720 |