Operator algebras over the p-adic integers
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910888984838144 |
|---|---|
| author | Buss, Alcides Garcia, Luiz Felipe Mukherjee, Devarshi |
| author_facet | Buss, Alcides Garcia, Luiz Felipe Mukherjee, Devarshi |
| contents | We introduce $p$-adic operator algebras, which are nonarchimedean analogues of $C^*$-algebras. We demonstrate that various classical examples of operator algebras - such as group(oid) $C^*$-algebras - have nonarchimedean counterparts. The category of $p$-adic operator algebras exhibits similar properties to those of the category of real and complex $C^*$-algebras, featuring limits, colimits, tensor products, crossed products and an enveloping construction permitting us to construct $p$-adic operator algebras from involutive algebras over $\mathbb{Z}_p$. In several cases of interest, the enveloping algebra construction recovers the $p$-adic completion of the underlying $\mathbb{Z}_p$-algebra. We then discuss an analogue of topological $K$-theory for Banach $\mathbb{Z}_p$-algebras, and compute it in basic examples such as the \(p\)-adic Cuntz algebra and rotation algebras. Finally, for a large class of $p$-adic operator algebras, we show that our $K$-theory coincides with the reduction mod $p$ of Quillen's algebraic $K$-theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04046 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Operator algebras over the p-adic integers Buss, Alcides Garcia, Luiz Felipe Mukherjee, Devarshi Operator Algebras 46L89, 46S10, 19D55 We introduce $p$-adic operator algebras, which are nonarchimedean analogues of $C^*$-algebras. We demonstrate that various classical examples of operator algebras - such as group(oid) $C^*$-algebras - have nonarchimedean counterparts. The category of $p$-adic operator algebras exhibits similar properties to those of the category of real and complex $C^*$-algebras, featuring limits, colimits, tensor products, crossed products and an enveloping construction permitting us to construct $p$-adic operator algebras from involutive algebras over $\mathbb{Z}_p$. In several cases of interest, the enveloping algebra construction recovers the $p$-adic completion of the underlying $\mathbb{Z}_p$-algebra. We then discuss an analogue of topological $K$-theory for Banach $\mathbb{Z}_p$-algebras, and compute it in basic examples such as the \(p\)-adic Cuntz algebra and rotation algebras. Finally, for a large class of $p$-adic operator algebras, we show that our $K$-theory coincides with the reduction mod $p$ of Quillen's algebraic $K$-theory. |
| title | Operator algebras over the p-adic integers |
| topic | Operator Algebras 46L89, 46S10, 19D55 |
| url | https://arxiv.org/abs/2403.04046 |