Simple C*-algebras with locally finite decomposition rank

TL;DR

This paper introduces locally finite decomposition rank for simple unital C*-algebras and demonstrates its significance in classifying nuclear C*-algebras, confirming the Elliott conjecture for certain classes.

Contribution

It defines the concept of locally finite decomposition rank and shows its implications for classification and structural properties of simple unital C*-algebras.

Findings

Simple unital C*-algebras with locally finite decomposition rank and Z-stability have tracial rank zero.

Such algebras are approximately homogeneous of topological dimension at most 3.

The results confirm the Elliott conjecture for Z-stable ASH algebras with real rank zero.

Abstract

We introduce the notion of locally finite decomposition rank, a structural property shared by many stably finite nuclear C*-algebras. The concept is particularly relevant for Elliott's program to classify nuclear C*-algebras by K-theory data. We study some of its properties and show that a simple unital C*-algebra, which has locally finite decomposition rank, real rank zero and which absorbs the Jiang-Su algebra Z tensorially, has tracial rank zero in the sense of Lin. As a consequence, any such C*-algebra, if it additionally satisfies the Universal Coefficients Theorem, is approximately homogeneous of topological dimension at most 3. Our result in particular confirms the Elliott conjecture for the class of simple unital Z-stable ASH algebras with real rank zero. Moreover, it implies that simple unital Z-stable AH algebras with real rank zero not only have slow dimension growth in the ASH sense, but even in the AH sense.