Strict comparison holds in the uniform Roe algebra of a discrete amenable group

TL;DR

This paper proves that in the uniform Roe algebra of a countable discrete amenable group, positive elements are Cuntz subequivalent if their traces satisfy a strict inequality.

Contribution

It establishes a strict comparison result for positive elements in the uniform Roe algebra of an amenable group, extending the understanding of Cuntz subequivalence.

Findings

Positive elements with smaller trace are Cuntz subequivalent to those with larger trace.

Strict comparison holds in the uniform Roe algebra of amenable groups.

The result applies to algebras formed from the universal minimal set of the group.

Abstract

Let $\Gamma$ be a countable discrete amenable group, and let $A=l^\infty(\Gamma) \rtimes \Gamma$ or $A = \mathrm{C}(M) \rtimes \Gamma$, where $(M, \Gamma)$ is the universal minimal set of $\Gamma$. It is shown that if $a, b \in A \otimes \mathcal K$ are positive elements such that $$\mathrm{d}_\tau(a) < \mathrm{d}_\tau(b),\quad \tau \in \mathrm{T}(A),$$ then $a$ is Cuntz subequivalent to $b$.