Full crossed products by Hopf C*-algebras

We show that when a co-involutive Hopf C*-algebra $S$ coacts via $\delta$ on a C*-algebra $A$, there exists a full crossed product $A{\times }_{\delta }S$, with universal properties analogous to those of full crossed products by locally compact groups. The dual Hopf C*-algebra is then defined by $\hat S:={\Bbb C}\times_{\id} S$.