A non-Archimedean counterpart of Johnson's theorem for discrete groups

TL;DR

This paper introduces a non-Archimedean analogue of Johnson's theorem, establishing a new form of K-amenability for discrete groups over spherically complete fields and linking it to the amenability of associated Banach algebras.

Contribution

It defines a novel concept of K-amenability for discrete groups over non-Archimedean fields and proves its equivalence to the amenability of the Banach algebra l^1(G).

Findings

K-amenability is characterized for discrete groups over spherically complete fields.

The Banach K-algebra l^1(G) is amenable iff G is K-amenable.

Establishes a non-Archimedean version of Johnson's theorem.

Abstract

Let $K$ be a spherically complete field with a non-Archimedean valuation. We define a new version of $K-$amenability for discrete groups and show that the Banach $K-$algebra $l^1(G)$ is amenable iff $G$ is $K-$amenable in our sense.