Divisibility and Real Rank Zero

TL;DR

This paper establishes the equivalence of several regularity properties for simple separable exact $C^*$-algebras with traces, and analyzes the structure of their tracial completions, with implications for classification.

Contribution

It proves the equivalence of multiple regularity conditions and characterizes the tracial completions of certain $C^*$-algebras, advancing the understanding of their structure and classification.

Findings

Equivalence of regularity properties for $A$ with traces.

Tracial completions are hyperfinite, type II$_1$, and isomorphic to the hyperfinite II$_1$ factor.

Simple unital diagonal AH-algebras satisfy tracial strict comparison.

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 $d_\tau(a)<d_\tau(b)$ holds for all traces $\tau\in\rT(V),$ then there is a sequence $\{r_n\}\subset V$ such that $\lim_n\|a-r_n^*br_n\|_{2,\rT(V)}=0.$