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

Qingnan An, Søren Eilers, Guihua Gong, Zhichao Liu

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.

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

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 algebra, when 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 -algebras of real rank zero and stable rank one.

Paper Structure

This paper contains 6 sections, 12 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The ASH construction
  4. Obstruction in quotients
  5. full extension case and classification
  6. The example with bounded torsion

Key Result

Theorem 1.1

(Theorem ex1 and Example ex torsion) There exists a real rank zero $C^*$-algebra $E$ which can be written as an inductive limit of 1-NCCW complexes but $E$ is not an A$\mathcal{HD}$ algebra. Such $E$ can have torsion free or bounded torsion $K_1$-group.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 24 more