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

Abstract

Let $\Gamma$ be a countable discrete amenable group, and let $A=l^{\infty }(\Gamma )⋊\Gamma$. 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 $(M,\Gamma )$. The simple C*-algebra $\mathrm{C}(M)⋊\Gamma$ 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, $\mathrm{C}(M)⋊\Gamma$ is approximately divisible.

27 pages. References are revised; Section 4 is extended on the C*-algebra of the universal minimal set