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A CRITERION FOR HYPERSYMMETRY ON DISCRETE GROUPOIDS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 2 / October 2023
- Published online by Cambridge University Press:
- 05 December 2022, pp. 320-330
- Print publication:
- October 2023
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- Article
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Given a Fell bundle
$\mathscr C\overset {q}{\to }\Xi $ over the discrete groupoid 
$\Xi $, we study the symmetry of the associated Hahn algebra 
$\ell ^{\infty ,1}(\Xi \!\mid \!\mathscr C)$ in terms of the isotropy subgroups of 
$\Xi $. We prove that 
$\Xi $ is symmetric (respectively hypersymmetric) if and only if all of the isotropy subgroups are symmetric (respectively hypersymmetric). We also characterise hypersymmetry using Fell bundles with constant fibres, showing that for discrete groupoids, ‘hypersymmetry’ equals ‘rigid symmetry’.
