Locally convex bialgebroid of an action Lie groupoid

Action Lie groupoids are used to model spaces of orbits of actions of Lie groups on manifolds. For each such action groupoid $M⋊H$ we construct a locally convex bialgebroid $(M⋊H)$ with an antipode over $(M)$, from which the groupoid $M⋊H$ can be reconstructed as its spectral action Lie groupoid $((M⋊H))$.