Inductive Limits of Subhomogeneous $C^*$-algebras with Hausdorff Spectrum

TL;DR

This paper studies inductive limits of subhomogeneous C*-algebras with Hausdorff spectrum, showing they have tracial rank at most one when tensorized with the rational UHF-algebra, leading to classification results based on Elliott invariants.

Contribution

It proves that certain inductive limits of generalized dimension drop C*-algebras have tracial rank at most one after tensoring with Q, enabling classification by Elliott invariants.

Findings

A⊗Q has tracial rank ≤ 1 for these algebras.

Classification of these algebras is complete via Elliott invariants.

Includes all unital simple AH-algebras and dimension drop C*-algebras.

Abstract

We consider unital simple inductive limits of generalized dimension drop C*-algebras They are so-called ASH-algebras and include all unital simple AH-algebras and all dimension drop $C^*$-algebras. Suppose that $A$ is one of these C*-algebras. We show that $A\otimes Q$ has tracial rank no more than one, where $Q$ is the rational UHF-algebra. As a consequence, we obtain the following classification result: Let $A$ and $B$ be two unital simple inductive limits of generalized dimension drop algebras with no dimension growth. Then $A\cong B$ if and only if they have the same Elliott invariant.