Coactions and Crossed Products of Hopf C*-Algebras

In this paper, we study coactions and their crossed products by Hopf C*-algebras defined by multiplicative unitaries. We generalize some results in the case of group actions and coactions. Furthermore, when applied to the group case, some of these generalizations improve the original results. For example, we obtained the following result. Let A</it> be a C*-algebra with coaction ɛ by S</it> where S</it> is a Hopf C*-algebra defined by an amenable multiplicative unitary. If A</it> is nuclear or C*-exact, then so is A</it> X<inf>ɛ, r</inf>Ŝ</it>. This implies that if (A</it>, G</it>, α</it>) is any C*-dynamical system such that A</it> X<inf>α, r</inf>G</it> is nuclear or C*-exact, then so is A</it>.