We study the relationship between antipodes on a Hopf algebroid H in the sense of B & ouml;hm-Szlachanyi and the group of twists that lies inside the associated convolution algebra. We specialize to the case of a faithfully flat H-Hopf-Galois extensions B subset of A and related Ehresmann-Schauenburg bialgebroid. In particular, we find that the twists are in one-toone correspondence with H-comodule algebra automorphism of A. We work out in detail the U(1) -extension O ( C P n - 1 q ) subset of O (S 2 n - 1 q ) on the quantum projective space and show how to get an antipode on the bialgebroid out of the K -theory of the base algebra O ( C P n - 1 q ). (c) 2024 Published by Elsevier Inc.
