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Operator algebras over the p-adic integers

Alcides Buss, Luiz Felipe Garcia, Devarshi Mukherjee

Abstract

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.

Operator algebras over the p-adic integers

Abstract

We introduce -adic operator algebras, which are nonarchimedean analogues of -algebras. We demonstrate that various classical examples of operator algebras - such as group(oid) -algebras - have nonarchimedean counterparts. The category of -adic operator algebras exhibits similar properties to those of the category of real and complex -algebras, featuring limits, colimits, tensor products, crossed products and an enveloping construction permitting us to construct -adic operator algebras from involutive algebras over . In several cases of interest, the enveloping algebra construction recovers the -adic completion of the underlying -algebra. We then discuss an analogue of topological -theory for Banach -algebras, and compute it in basic examples such as the -adic Cuntz algebra and rotation algebras. Finally, for a large class of -adic operator algebras, we show that our -theory coincides with the reduction mod of Quillen's algebraic -theory.
Paper Structure (23 sections, 61 theorems, 249 equations)

This paper contains 23 sections, 61 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on nonarchimedean analysis
  3. The $p$-adic analogue of a Hilbert space
  4. Bounded operators on $p$-adic Hilbert spaces
  5. Matrices of Operators
  6. Continuous functional calculus
  7. $p$-adic Operator Algebras
  8. A remark on involutions
  9. The category of $p$-adic operator algebras
  10. Limits in $\mathrm{OpeAlg}_p$
  11. Enveloping $p$-adic operator algebra and colimits in $\mathrm{OpeAlg}_p$
  12. Tensor products
  13. The maximal tensor product
  14. The spatial tensor product
  15. Crossed Products
  16. ...and 8 more sections

Key Result

Theorem 1.1

The category of $p$-adic operator algebras and contractive $*$-algebra homomorphisms has all limits and colimits.

Theorems & Definitions (152)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 3.1: The space $\mathbb{Q}_{p}(X)$
  • ...and 142 more