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Continuous fields of interval algebras

Laurent Cantier

TL;DR

The paper addresses the classification of one-parameter continuous fields of unital AI-algebras by extending Cuntz semigroup methods to the field setting. It develops a local-to-global framework that combines Michael's selection theorem with a novel gluing technique to lift fiberwise Cu-data to continuous-field morphisms, and introduces a field version of the Cu-category to obtain a functorial viewpoint. For a natural inductive-limit class $\mathcal{C}$ built from $C([0,1])$ tensored with unital interval algebras, the authors prove existence and uniqueness results for morphisms, and show that Cu-isomorphisms descending to fibers lift to field isomorphisms, yielding a Cu-based classification (CP). The work thus provides a constructive, fiber-aware Cu-classification in the AI-interval regime and lays groundwork for a Cu-functorial framework of continuous fields, with potential to distinguish and relate such fields via field-level invariants.

Abstract

This paper investigates and classifies a specific class of one-parameter continuous fields of C*-algebras, which can be seen as generalized AI-algebras. Building on the classification of *-homomorphisms between interval algebras by the Cuntz semigroup, along with a selection theorem and a gluing procedure, we employ a 'local-to-global' strategy to achieve our classification result.

Continuous fields of interval algebras

TL;DR

The paper addresses the classification of one-parameter continuous fields of unital AI-algebras by extending Cuntz semigroup methods to the field setting. It develops a local-to-global framework that combines Michael's selection theorem with a novel gluing technique to lift fiberwise Cu-data to continuous-field morphisms, and introduces a field version of the Cu-category to obtain a functorial viewpoint. For a natural inductive-limit class built from tensored with unital interval algebras, the authors prove existence and uniqueness results for morphisms, and show that Cu-isomorphisms descending to fibers lift to field isomorphisms, yielding a Cu-based classification (CP). The work thus provides a constructive, fiber-aware Cu-classification in the AI-interval regime and lays groundwork for a Cu-functorial framework of continuous fields, with potential to distinguish and relate such fields via field-level invariants.

Abstract

This paper investigates and classifies a specific class of one-parameter continuous fields of C*-algebras, which can be seen as generalized AI-algebras. Building on the classification of *-homomorphisms between interval algebras by the Cuntz semigroup, along with a selection theorem and a gluing procedure, we employ a 'local-to-global' strategy to achieve our classification result.

Paper Structure

This paper contains 7 sections, 23 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. An intro on continuous fields of $C^*$-algebras
  3. From local to global
  4. The existence
  5. The uniqueness
  6. The classification
  7. An outro on continuous fields of Cuntz semigroups

Key Result

Theorem 1

Let $\mathcal{A},\mathcal{B}\in \mathcal{C}$. Let $\alpha\colon \mathop{\mathrm{Cu}}\nolimits(\mathcal{A})\rightarrow \mathop{\mathrm{Cu}}\nolimits(\mathcal{B})$ be any scaled $\mathop{\mathrm{Cu}}\nolimits$-morphism descending to the fibers. Then there exists a continuous field morphism $\phi\colon

Theorems & Definitions (52)

  • Theorem
  • Definition 1
  • Proposition 1: Continuity Criterion
  • proof
  • Remark 1
  • Definition 2
  • Proposition 2
  • proof
  • Theorem 1
  • Definition 3
  • ...and 42 more