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Operator algebras over the p-adic integers -- II

Alcides Buss, Luiz Felipe Garcia, Devarshi Mukherjee

TL;DR

This work develops a foundational framework for operator algebras over the $p$-adic integers, introducing $p$-adic von Neumann algebras and a $p$-adic GNS theory to generate broad classes of examples, including residually finite-rank and affinoid algebras. It characterizes centers and simplicity in the $p$-adic setting, establishes a $p$-adic analogue of GNS representations, and analyzes $K$-theory for both commutative and noncommutative $p$-adic operator algebras, linking homotopy analytic and continuous $K$-theories via descent formalisms. The results extend the foundations of the emerging $p$-adic operator algebra framework and connect noncommutative geometry with $p$-adic analytic geometry. Overall, the paper provides structural, representation-theoretic, and $K$-theoretic tools to study and construct rich families of $p$-adic operator algebras.

Abstract

We continue the study of operator algebras over the $p$-adic integers, initiated in our previous work [1]. In this sequel, we develop further structural results and provide new families of examples. We introduce the notion of $p$-adic von Neumann algebras, and analyze those with trivial center, that we call ''factors''. In particular we show that ICC groups provide examples of factors. We then establish a characterization of $p$-simplicity for groupoid operator algebras, showing its relation to effectiveness and minimality. A central part of the paper is devoted to a $p$-adic analogue of the GNS construction, leading to a representation theorem for Banach $^*$-algebras over $\mathbb{Z}_p$. As applications, we exhibit large classes of $p$-adic operator algebras, including residually finite-rank algebras and affinoid algebras with the spectral norm. Finally, we investigate the $K$-theory of $p$-adic operator algebras, including the computation of homotopy analytic $K$-theory of continuous $\mathbb{Z}_p$-valued functions on a compact Hausdorff space and the analytic (non-homotopy invariant) $K$-theory of certain $p$-adically complete Banach algebras in terms of continuous $K$-theory. Together, these results extend the foundations of the emerging theory of $p$-adic operator algebras.

Operator algebras over the p-adic integers -- II

TL;DR

This work develops a foundational framework for operator algebras over the -adic integers, introducing -adic von Neumann algebras and a -adic GNS theory to generate broad classes of examples, including residually finite-rank and affinoid algebras. It characterizes centers and simplicity in the -adic setting, establishes a -adic analogue of GNS representations, and analyzes -theory for both commutative and noncommutative -adic operator algebras, linking homotopy analytic and continuous -theories via descent formalisms. The results extend the foundations of the emerging -adic operator algebra framework and connect noncommutative geometry with -adic analytic geometry. Overall, the paper provides structural, representation-theoretic, and -theoretic tools to study and construct rich families of -adic operator algebras.

Abstract

We continue the study of operator algebras over the -adic integers, initiated in our previous work [1]. In this sequel, we develop further structural results and provide new families of examples. We introduce the notion of -adic von Neumann algebras, and analyze those with trivial center, that we call ''factors''. In particular we show that ICC groups provide examples of factors. We then establish a characterization of -simplicity for groupoid operator algebras, showing its relation to effectiveness and minimality. A central part of the paper is devoted to a -adic analogue of the GNS construction, leading to a representation theorem for Banach -algebras over . As applications, we exhibit large classes of -adic operator algebras, including residually finite-rank algebras and affinoid algebras with the spectral norm. Finally, we investigate the -theory of -adic operator algebras, including the computation of homotopy analytic -theory of continuous -valued functions on a compact Hausdorff space and the analytic (non-homotopy invariant) -theory of certain -adically complete Banach algebras in terms of continuous -theory. Together, these results extend the foundations of the emerging theory of -adic operator algebras.

Paper Structure

This paper contains 9 sections, 31 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. $p$-adic von Neumann algebras
  3. Centers and factor $p$-adic von Neumann algebras
  4. $p$-Simplicity
  5. A $p$-adic GNS Theorem
  6. Finite-rank quasi-Hilbert spaces
  7. Residually finite-rank Banach $*$-algebras
  8. Homotopy analytic $K$-theory for commutative $p$-adic operator algebras
  9. Non-homotopy invariant $K$-theory for $p$-adic operator algebras

Key Result

Proposition 2.4

Let $\{M_i : i \in I\}$ be a family of $p$-adic von Neumann algebras acting on the same $p$-adic Hilbert space $\mathcal{H}=\mathbb{Q}_{p}(X)$. Then their intersection is again a $p$-adic von Neumann algebra on $\mathcal{H}$; equivalently $M"=M$.

Theorems & Definitions (81)

  • Definition 2.1: von Neumann $p$-adic operator algebra
  • Remark 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Remark 2.6
  • Example 2.7
  • Remark 2.8
  • Proposition 2.9
  • ...and 71 more