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Towards a classification of unitary elements of C*-algebras

Laurent Cantier

TL;DR

This work studies the classification of unitary elements in unital $C^*$-algebras through the Cuntz semigroup $\mathrm{Cu}$. It develops a framework using a revised Cu-functor and weakly semiprojective Cu-morphisms to address the existence problem, proving complete results for unital AF algebras and extending to AH$_1$-algebras. The paper confirms that $\mathrm{Cu}$ robustly handles existence in AF cases, but demonstrates obstructions to uniqueness beyond AF, even in real rank zero, and shows notable obstructions in real rank one (e.g., in $\mathcal{Z}$). It argues that additional $\mathrm{K}_1$-like information, via a Hausdorffized or unitary-enhanced invariant, is needed to achieve full classification beyond AF. These results point toward a refined invariant that combines $\mathrm{Cu}$ with $\overline{\mathrm{K}}^{\mathrm{alg}}_1$-data to classify unitary elements in broader AH$_1$-and AT-structured algebras.

Abstract

In [5] the author conjectures and partially shows that the Cuntz semigroup classifies unitary elements of unital AF-algebras. We provide a complete proof by addressing the existence part of the conjecture, under a mild adjustment of both domain and codomain of the functor Cu. We also tackle the classification beyond the AF case and more particularly, we look at unitary elements of what we call AH$_1$-algebras. We obtain positive progress as far as the existence part is concerned. Nevertheless, we reveal that extra information is needed for the uniqueness part of the classification that the Cuntz semigroup fails to capture.

Towards a classification of unitary elements of C*-algebras

TL;DR

This work studies the classification of unitary elements in unital -algebras through the Cuntz semigroup . It develops a framework using a revised Cu-functor and weakly semiprojective Cu-morphisms to address the existence problem, proving complete results for unital AF algebras and extending to AH-algebras. The paper confirms that robustly handles existence in AF cases, but demonstrates obstructions to uniqueness beyond AF, even in real rank zero, and shows notable obstructions in real rank one (e.g., in ). It argues that additional -like information, via a Hausdorffized or unitary-enhanced invariant, is needed to achieve full classification beyond AF. These results point toward a refined invariant that combines with -data to classify unitary elements in broader AH-and AT-structured algebras.

Abstract

In [5] the author conjectures and partially shows that the Cuntz semigroup classifies unitary elements of unital AF-algebras. We provide a complete proof by addressing the existence part of the conjecture, under a mild adjustment of both domain and codomain of the functor Cu. We also tackle the classification beyond the AF case and more particularly, we look at unitary elements of what we call AH-algebras. We obtain positive progress as far as the existence part is concerned. Nevertheless, we reveal that extra information is needed for the uniqueness part of the classification that the Cuntz semigroup fails to capture.
Paper Structure (10 sections, 24 theorems, 44 equations)

This paper contains 10 sections, 24 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unitary Elements of AF algebras
  4. Unitary Elements of $\mathop{\mathrm{AH}}\nolimits_1$-algebras
  5. Obstruction to Uniqueness
  6. On the path towards Existence
  7. Real rank zero and beyond
  8. Real rank zero - On the path towards Uniqueness
  9. Real rank one - Obstruction to Uniqueness
  10. Closing Remark

Key Result

Theorem 1

Let $A$ be a unital $\mathop{\mathrm{AF}}\nolimits$-algebra. Let $\alpha:\mathop{\mathrm{Lsc}}\nolimits(\mathbb{T},\overline{\mathbb{N}})\longrightarrow \mathop{\mathrm{Cu}}\nolimits(A)$ be a weakly semiprojective $\mathop{\mathrm{Cu}}\nolimits$-morphism with $\alpha(1_\mathbb{T})=[1_A]$. Then there

Theorems & Definitions (45)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Definition 1
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • ...and 35 more