In this paper, we will prove that if A is a C-algebra with an effective coaction ǫ by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac C-algebras.

We construct the crossed product of a C(X)-algebra by an endomorphism, in such a way that it becomes induced by a Hilbert C(X)-bimodule. Furthermore we introduce the notion of C(X)-category, and discuss relationships with crossed products and duality for compact groups.

Let H be a finite Hopf C*-algebra and A of dimension. In this paper, we focus on the crossed product A⋊H arising from action A, which is ∗-algebra. terms faithful positive Haar measure C*-algebra, one can construct linear functional ∗-algebra A⋊H, further functional. Here, complete positivity plays vital role in argument. At last, conclude that dimension according to ∗- representation.