Divisibility and Real Rank Zero

Abstract

Let $A$ be a simple separable exact $C^*$-algebra that has traces. We show the following existed regularity properties are equivalent: \quad(1) $l^\infty(A)/J_A$ has real rank zero, where $J_A$ is the trace kernel ideal. \quad(2) $A$ is tracially almost divisible. \quad(3) $A$ is tracially $m$-almost divisible for some $m\in\N\cup\{0\}.$ \quad(4) $A$ has tracial approximate oscillation zero. \quad(5) $A$ has Property (TM). We also show that for an algebraically simple separable stable rank one \CA\ $B$ with non-empty compact ${\rm T}(B)$ and locally finite nuclear dimension, its uniform tracial completion $(\ol B^{\rT(B)}, \rT(B))$ is hyperfinite, type ${\rm II_1},$ and isomorphic to $({\cal R}_{\rT(B)},\rT(B))$. Furthermore, $\ol{B}^{{\rm T}(B)}$ is pure, has real rank zero and stable rank one, and satisfies $\rT (\ol B^{\rT(B)} )= \rT(B).$ Consequently, every simple separable unital diagonal AH-algebra $V$ (e.g. Villadsen algebras of the first type) has the following tracial strict comparison: For every $a,b\in V_+,$ if $d_τ(a)<d_τ(b)$ holds for all traces $τ\in\rT(V),$ then there is a sequence $\{r_n\}\subset V$ such that $\lim_n\|a-r_n^*br_n\|_{2,\rT(V)}=0.$