On the classification of simple amenable C*-algebras with finite decomposition rank

TL;DR

This paper classifies certain simple, amenable C*-algebras with finite decomposition rank, showing they are rationally AH algebras under specific conditions including Jiang-Su stability and K-theoretic properties.

Contribution

It establishes that unital simple separable C*-algebras with finite decomposition rank, Jiang-Su stability, and rational K_0-group are classified as rationally AH algebras.

Findings

Such algebras are ASH and rationally AH.

Finite decomposition rank combined with Jiang-Su stability implies classification.

K_0 tensor Q is isomorphic to Q for these algebras.

Abstract

Let $A$ be a unital simple separable C*-algebra satisfying the UCT. Assume that $\mathrm{dr}(A)<+\infty$, $A$ is Jiang-Su stable, and $\mathrm{K}_0(A)\otimes \mathbb{Q}\cong \mathbb{Q}$. Then $A$ is an ASH algebra (indeed, $A$ is a rationally AH algebra).