A short return to simple AH-algebras with real rank zero

TL;DR

This paper proves that certain unital simple AH-algebras with specific rank and torsion properties have tracial rank zero and are isomorphic to AH-algebras with no dimension growth, simplifying their classification.

Contribution

It establishes that under given conditions, these AH-algebras have tracial rank zero and are isomorphic to no-dimension-growth AH-algebras, extending classification results.

Findings

A unital simple AH-algebra with specified properties has tracial rank zero.

Such an algebra is isomorphic to a unital simple AH-algebra with no dimension growth.

The result applies to algebras with torsion in K_0 and real rank zero.

Abstract

Let $A$ be a unital simple AH-algebra with stable rank one and real rank zero such that $kx=0$ for all $x\in {\rm ker}\rho_A,$ the subgroup of infinitesmal elements in $K_0(A),$ and for the same integer $k\ge 1.$ We show that $A$ has tracial rank zero and is isomorphic to a unital simple AH-algebra with no dimension growth.