Braided module and comodule algebras, Galois extensions and elements of trace 1

Let k be a field and let H be a rigid braided Hopf k-algebra. In this paper we continue the study of the theory of braided Hopf crossed products began in [J.A. Guccione, J.J. Guccione, Theory of braided Hopf crossed products, J. Algebra 261 (2003) 54–101]. First we show that to have an H-braided comodule algebra is the same that to have an H†-braided module algebra, where H† is a variant of H∗, and then we study the maps and , that appear in the Morita context introduced in the above cited paper.