p进整数上的算子代数
Operator algebras over the p-adic integers
AI总结：
本研究引入C*-代数的非阿基米德类比——p进算子代数，证明其范畴具备与实/复C*-代数范畴相似的性质，构建了对应包络构造，讨论了Banach ℤ_p代数的拓扑K-理论类比并完成实例计算，验证了其与Quillen代数K-理论模p约化的一致性。
摘要：
我们引入p进算子代数，它是C*-代数的非阿基米德类比。我们证明了多种经典算子代数实例——例如群（群胚）C*-代数——都存在非阿基米德对应物。p进算子代数范畴展现出与实C*-代数、复C*-代数范畴相似的性质，具备极限、余极限、张量积、交叉积，以及一种包络构造，使我们能够从ℤ_p上的对合代数构造p进算子代数。在若干感兴趣的情形中，包络代数构造可恢复底层ℤ_p代数的p进完备化。随后我们讨论了Banach ℤ_p代数的拓扑K-理论类比，并在p进Cuntz代数、旋转代数等基础实例中完成了计算。最后，针对一大类p进算子代数，我们证明该K-理论与Quillen代数K-理论的模p约化相一致。
英文摘要：
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.
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