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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^*\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras. Along the way, we reprove some results of Kwaśniewski–Meyer on Fell bundle ideals.</p>

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Exactness and Fell bundles with the approximation property over inverse semigroups

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月28日

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  • Changyuan Gao,
  • Julian Kranz

摘要

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 \(C^*\) C ∗ -algebras. Along the way, we reprove some results of Kwaśniewski–Meyer on Fell bundle ideals.