2024/03/06 by Alcides Buss, Buss, Alcides, Luiz Felipe Garcia +3 · 1 citation
Mathematics · #19D55 #46L89 #46S10 #Advanced Algebra and Geometry #FOS: Mathematics #Mathematical and Theoretical Analysis #Operator Algebras (math.OA) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2403.04046
openalex publication_date 2024/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℤp. In several cases of interest, the enveloping algebra construction recovers the p-adic completion of the underlying ℤp-algebra. We then discuss an analogue of topological K-theory for Banach ℤp-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.