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

Let $\mathrm{\Gamma }$ be a countable discrete amenable group, and let $A=l^{\mathrm{\infty }}(\mathrm{\Gamma })⋊\mathrm{\Gamma }$. It is shown that if $a,b\in A\otimes \mathcal{K}$ are positive elements such that $${\mathrm{d}}_{\tau }(a)<{\mathrm{d}}_{\tau }(b),\tau \in \mathrm{T}(A),$$ then $a$ is Cuntz subequivalent to $b$. Moreover, consider the universal minimal set $(M,\mathrm{\Gamma })$. The simple C*-algebra $\mathrm{C}(M)⋊\mathrm{\Gamma }$ is shown to be AH in the strong sense that there is an increasing net of unital sub-C*-algebras $A_{\lambda }\subseteq A$, $\lambda \in \mathrm{\Lambda }$, such that each $A_{\lambda }$ is a simple (separable) $\mathcal{Z}$-absorbing approximately homogeneous C*-algebra with real rank zero and $A={\bigcup }_{\lambda \in \mathrm{\Lambda }}{A}_{\lambda }$. In particular, $\mathrm{C}(M)⋊\mathrm{\Gamma }$ is approximately divisible.