We consider the relationship between the Haagerup constants for a C-algebra and its crossed product by an amenable group. We prove equality when the group is discrete, and deduce equality when the group is compact and abelian.

in this paper, we presented algebroids and crossed modules of algebroids. we also presented pullback crossed module of algebroids and we then define pushout crossed modules of algebroids.

The notions of the crossed product and the braided category are discussed within the theory of C∗-algebras. Like in the purely algebraic approach of S. Majid these notions are used to generalize the quantum group theory. We give an example of a braided group. It coincides with the closed subset C = {z : |z| = . . . , q−2, q−1, 1, q, q2, . . . , 0} of the complex plane equipped with the q2braidi...