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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 Abe a C*‐algebra with coaction ɛ by Swhere Sis a Hopf C*‐algebra defined by an amenable multiplicative unitary. If Ais nuclear or C*‐exact, then so is AXɛ, rŜ. This implies that if (A, G, α) is any C*‐dynamical system such that AXα, rGis nuclear or C*‐exact, then so is A.

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 be a C*-algebra with coaction ɛ by S where S is a Hopf C*-algebra defined by an amenable multiplicative unitary. If A is nuclear or C*-exact, then so is A Xɛ, rŜ. This implies that if (A, G, α) is any C*-dynamical system such that A Xα, rG is nuclear or C*-exact, then so is A.