# Aspects of $p$-adic operator algebras

## TL;DR

This paper introduces a $p$-adic analogue of Hilbert spaces, explores related operator algebras, computes their $K$-theory, and surveys foundational results, extending functional analysis into the $p$-adic setting.

## Contribution

It develops a $p$-adic framework for Hilbert spaces and operator algebras, including $K$-theory calculations, which is a novel extension of classical analysis.

## Key findings

- Computed $K$-theory of $p$-adic compact operators
- Analyzed properties of $p$-adic bounded operators
- Surveyed foundational results in $p$-adic operator theory

## Abstract

In this article, we propose a $p$-adic analogue of complex Hilbert space and consider generalizations of some well-known theorems from functional analysis and the basic study of operators on Hilbert spaces. We compute the $K$-theory of the analogue of the algebra of compact operators and the algebra of all bounded operators. This article contains a survey on results from the thesis of the first author.