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Exact Fell bundles with the approximation property over inverse semigroups

Changyuan Gao, Julian Kranz

TL;DR

This work shows that for a Fell bundle $\mathcal{A}$ over a unital inverse semigroup with the approximation property, the reduced cross-sectional algebra $C^*_r(\mathcal{A})$ is exact precisely when the unit fiber $A_1$ is exact, extending known dynamical results. The argument leverages the approximation property to transfer exactness from the unit fiber to the cross-sectional algebra via tensoring, after establishing that Fell bundle ideals correspond to invariant ideals and that short exact sequences of bundles yield exact sequences at the level of full $C^*$-algebras. A key technical step is the compatibility of the left-regular construction with tensor products, giving $C^*_r(\mathcal{A}\otimes D)\cong C^*_r(\mathcal{A})\otimes D$. The results remove separability and Hausdorff assumptions and apply to étale groupoids via associated Fell bundles, with Kwaśniewski–Meyer’s ideal theory revisited in detail for completeness.

Abstract

We prove that the reduced cross-sectional algebra of a Fell bundle with the approximation property over an inverse semigroup is exact if and only if the unit fiber of the Fell bundle is exact. This generalizes a recent result of the first-named author for actions of second countable locally compact Hausdorff groupoids on separable $C^*$-algebras. Along the way, we reprove some results of Kwaśniewski--Meyer on Fell bundle ideals.

Exact Fell bundles with the approximation property over inverse semigroups

TL;DR

This work shows that for a Fell bundle over a unital inverse semigroup with the approximation property, the reduced cross-sectional algebra is exact precisely when the unit fiber is exact, extending known dynamical results. The argument leverages the approximation property to transfer exactness from the unit fiber to the cross-sectional algebra via tensoring, after establishing that Fell bundle ideals correspond to invariant ideals and that short exact sequences of bundles yield exact sequences at the level of full -algebras. A key technical step is the compatibility of the left-regular construction with tensor products, giving . The results remove separability and Hausdorff assumptions and apply to étale groupoids via associated Fell bundles, with Kwaśniewski–Meyer’s ideal theory revisited in detail for completeness.

Abstract

We prove that the reduced cross-sectional algebra of a Fell bundle with the approximation property over an inverse semigroup is exact if and only if the unit fiber of the Fell bundle is exact. This generalizes a recent result of the first-named author for actions of second countable locally compact Hausdorff groupoids on separable -algebras. Along the way, we reprove some results of Kwaśniewski--Meyer on Fell bundle ideals.
Paper Structure (4 sections, 13 theorems, 54 equations)

This paper contains 4 sections, 13 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ideals in Fell bundles
  4. Proof of the main result

Key Result

Theorem A

Let $\mathcal{A}$ be a Fell bundle over a unital inverse semigroup $S$ with the approximation property. Then the reduced cross-sectional algebra $C^*_r(\mathcal{A})$ is exact if and only if its unit fiber $A_1$ is exact.

Theorems & Definitions (35)

  • Theorem A
  • Corollary 1
  • Definition 1: BM-17
  • Theorem 2.1: BM-17
  • Remark 1
  • Definition 2: BEM-17
  • Definition 3: BEM-17
  • Proposition 1: BEM-17
  • Remark 2
  • Definition 4: BM-23
  • ...and 25 more