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Co-universal $C^{\ast}$-algebras for product systems over finite aligned subcategories of groupoids

Feifei Miao, Liguang Wang, Wei Yuan

TL;DR

This work develops a framework to study co-universal C*-algebras for compactly aligned product systems over finite aligned subcategories of groupoids. By extending product systems to left cancellative small categories, introducing compact alignment, and formulating Nica covariant Toeplitz representations, the authors construct a Nica-Toeplitz apparatus and relate it to cosystems and Fell bundles. They prove the existence of a co-universal C*-algebra for injective, gauge-compatible, Nica covariant representations, equipped with a normal coaction of the underlying groupoid; this C*-envelope is realized via cross-sectional algebras of Fell bundles and the coaction theory of cosystems. The results generalize prior co-universal phenomena from groups and right LCM semigroups to groupoid subcategories, providing a robust, scalable method to obtain co-universal objects and their normal coactions in a groupoid-graded setting.

Abstract

The product systems over left cancellative small categories are introduced and studied in this paper. We also introduce the notion of compactly aligned product systems over finite aligned left cancellative small categories and its Nica covariant representations. The existence of co-universal algebras for injective, gauge-compatible, Nica covariant representations of compactly aligned product systems over finite aligned subcategories of groupoids is proved in this paper.

Co-universal $C^{\ast}$-algebras for product systems over finite aligned subcategories of groupoids

TL;DR

This work develops a framework to study co-universal C*-algebras for compactly aligned product systems over finite aligned subcategories of groupoids. By extending product systems to left cancellative small categories, introducing compact alignment, and formulating Nica covariant Toeplitz representations, the authors construct a Nica-Toeplitz apparatus and relate it to cosystems and Fell bundles. They prove the existence of a co-universal C*-algebra for injective, gauge-compatible, Nica covariant representations, equipped with a normal coaction of the underlying groupoid; this C*-envelope is realized via cross-sectional algebras of Fell bundles and the coaction theory of cosystems. The results generalize prior co-universal phenomena from groups and right LCM semigroups to groupoid subcategories, providing a robust, scalable method to obtain co-universal objects and their normal coactions in a groupoid-graded setting.

Abstract

The product systems over left cancellative small categories are introduced and studied in this paper. We also introduce the notion of compactly aligned product systems over finite aligned left cancellative small categories and its Nica covariant representations. The existence of co-universal algebras for injective, gauge-compatible, Nica covariant representations of compactly aligned product systems over finite aligned subcategories of groupoids is proved in this paper.
Paper Structure (8 sections, 19 theorems, 100 equations)

This paper contains 8 sections, 19 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Quasi-product systems over left cancellative small categories
  3. Product systems over left cancellative categories and their Toeplitz representations
  4. Compactly aligned product systems over finite aligned left cancellative categories
  5. The Nica covariant representation of compactly aligned product systems
  6. Co-systems and their $C^{\ast}$-envelopes
  7. Fell bundles over groupoids and C$^*$-algebras graded by groupoids
  8. $C^*$-envelopes of cosystems as co-universal $C^*$-algebras

Key Result

Theorem 1

Let $\mathfrak{X} = (\{\EuScript{A}_{x}\}, \{\mathfrak{X}_{f}\})$ be a compactly aligned product system over a finite aligned subcategory of a groupoid $G$. Then the co-universal C$^*$-algebra for injective, gauge-compatible, Nica covariant representations of $\mathfrak{X}$ exists. Moreover, the coa

Theorems & Definitions (68)

  • Theorem 1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Definition 2.4
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5: MR3909245
  • ...and 58 more