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

Abstract

Let $\Gamma$ be a countable discrete amenable group, and let $A=l^\infty(\Gamma) \rtimes \Gamma$. 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 $b$. Moreover, consider the universal minimal set $(M, \Gamma)$. The simple C*-algebra $\mathrm{C}(M)\rtimes\Gamma$ is shown to be AH in the strong sense that there is an increasing net of unital sub-C*-algebras $A_\lambda \subseteq A$, $\lambda \in \Lambda$, such that each $A_\lambda$ is a simple (separable) $\mathcal Z$-absorbing approximately homogeneous C*-algebra with real rank zero and $A = \bigcup_{\lambda \in \Lambda} A_\lambda$. In particular, $\mathrm{C}(M)\rtimes\Gamma$ is approximately divisible.