Operator algebras over the p-adic integers

We introduce $p$-adic operator algebras, which are nonarchimedean analogues of $C^{\ast }$-algebras. We demonstrate that various classical examples of operator algebras - such as group(oid) $C^{\ast }$-algebras - have nonarchimedean counterparts. The category of $p$-adic operator algebras exhibits similar properties to those of the category of real and complex $C^{\ast }$-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.