“…This procedure has been introduced independently, and under different forms, by Connes and Moscovici [7] in their study of the index theory of transversely elliptic operators, and by Kadison [19] in connection with his work on (pseudo-)Galois extensions. Later, Panaite and Van Oystaeyen proved in [31] that the two constructions were in fact equivalent (isomorphic as A-bialgebroids) and that, as algebras, they were particular instances of the L-R-smash product introduced in [30]. Nevertheless, by following [4], we will refer to the bialgebroid A ⊙ H ⊙ A as the Connes-Moscovici's bialgebroid.…”