A Note On Subhomogeneous C*-Algebras

TL;DR

This paper proves that finitely generated subhomogeneous C*-algebras have finite decomposition rank, leading to new classifications of certain ASH algebras and confirming aspects of the Elliott conjecture.

Contribution

It establishes that finitely generated subhomogeneous C*-algebras have finite decomposition rank, enabling classification results for a broad class of ASH algebras.

Findings

Finitely generated subhomogeneous C*-algebras have finite decomposition rank.

Separable ASH C*-algebras can be expressed as inductive limits of such algebras.

Certain simple unital ASH algebras satisfy the Elliott conjecture.

Abstract

We show that finitely generated subhomogeneous C*-algebras have finite decomposition rank. As a consequence, any separable ASH C*-algebra can be written as an inductive limit of subhomogeneous C*-algebras each of which has finite decomposition rank. It then follows from work of H. Lin and of the second named author that the class of simple unital ASH algebras which have real rank zero and absorb the Jiang-Su algebra tensorially satisfies the Elliott conjecture.