Morita equivalences between fixed point algebras and crossed products

In this paper, we will prove that if $A$ is a $C^*$-algebra with an effective coaction $\epsilon$ 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.