Subhomogeneity in the classification of real rank zero C*-algebras

TL;DR

This paper explores the limitations of K-theory invariants in classifying real rank zero C*-algebras, constructing examples that challenge existing classification assumptions and revealing obstructions in the invariants.

Contribution

It constructs new classes of ASH algebras that are not K-pure and demonstrates that total K-theory is insufficient for classification, highlighting obstructions in the process.

Findings

Existence of non-K-pure ASH algebras of real rank zero.

Total K-theory does not fully classify these algebras.

Obstructions in K-theory of ideals and quotients affect classification.

Abstract

In this paper, we construct a class of ASH algebras of real rank zero and stable rank one which is not K-pure. Then we show the following: (i) There exists a real rank zero inductive limit of 1-dimensional noncommutative CW complexes which is not an A$\mathcal{HD}$ algebra, when $K_1$ is torsion free or has bounded torsion. (ii) Total K-theory is not a complete invariant for ASH algebras of real rank zero. (iii) There are obstructions both in the total K-theory of ideals and quotients in the classification of $C^*$-algebras of real rank zero and stable rank one.