Amenability of coarse spaces and K-algebras

TL;DR

This paper explores the concept of amenability in coarse spaces and algebras, establishing connections between geometric and algebraic properties, and applying these ideas to specific algebra classes like Leavitt path algebras.

Contribution

It introduces an algebraic perspective on amenability and paradoxical decompositions, linking metric space properties with algebraic amenability in translation algebras.

Findings

Amenability of metric spaces is equivalent to algebraic amenability of their translation algebras.

The paper characterizes amenability and paradoxical decompositions in algebraic terms.

It applies the theory to Leavitt path algebras and translation algebras, demonstrating practical implications.

Abstract

In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over commutative fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra.