OPERATOR ALGEBRAS OVER THE p-ADIC INTEGERS

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