Computable $K$-theory for $\mathrm{C}^*$-algebras: UHF algebras
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| author | Eagle, Christopher Goldbring, Isaac McNicholl, Timothy Miller, Russell |
| author_facet | Eagle, Christopher Goldbring, Isaac McNicholl, Timothy Miller, Russell |
| contents | We initiate the study of the effective content of $K$-theory for $\mathrm{C}^*$-algebras. We prove that there are computable functors which associate, to a computably enumerable presentation of a $\mathrm{C}^*$-algebra $\boldA$, computably enumerable presentations of the abelian groups $K_0(\boldA)$ and $K_1(\boldA)$. When $\boldA$ is stably finite, we show that the positive cone of $K_0(\boldA)$ is computably enumerable. We strengthen the results in the case that $\boldA$ is a UHF algebra by showing that the aforementioned presentation of $K_0(\boldA)$ is actually computable. In the UHF case, we also show that $\boldA$ has a computable presentation precisely when $K_0(\boldA)$ has a computable presentation, which in turn is equivalent to the supernatural number of $\boldA$ being lower semicomputable; we give an example that shows that this latter equivalence cannot be improved to requiring that the supernatural number of $\boldA$ is computable. Finally, we prove that every UHF algebra is computably categorical. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08526 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computable $K$-theory for $\mathrm{C}^*$-algebras: UHF algebras Eagle, Christopher Goldbring, Isaac McNicholl, Timothy Miller, Russell Logic Operator Algebras 03D78 We initiate the study of the effective content of $K$-theory for $\mathrm{C}^*$-algebras. We prove that there are computable functors which associate, to a computably enumerable presentation of a $\mathrm{C}^*$-algebra $\boldA$, computably enumerable presentations of the abelian groups $K_0(\boldA)$ and $K_1(\boldA)$. When $\boldA$ is stably finite, we show that the positive cone of $K_0(\boldA)$ is computably enumerable. We strengthen the results in the case that $\boldA$ is a UHF algebra by showing that the aforementioned presentation of $K_0(\boldA)$ is actually computable. In the UHF case, we also show that $\boldA$ has a computable presentation precisely when $K_0(\boldA)$ has a computable presentation, which in turn is equivalent to the supernatural number of $\boldA$ being lower semicomputable; we give an example that shows that this latter equivalence cannot be improved to requiring that the supernatural number of $\boldA$ is computable. Finally, we prove that every UHF algebra is computably categorical. |
| title | Computable $K$-theory for $\mathrm{C}^*$-algebras: UHF algebras |
| topic | Logic Operator Algebras 03D78 |
| url | https://arxiv.org/abs/2501.08526 |