A CRITERION FOR HYPERSYMMETRY ON DISCRETE GROUPOIDS.

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'. [ABSTRACT FROM AUTHOR]