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An equivariant completely bounded linear operator between two C*-algebras acted on by an amenable group is shown to lift to a completely bounded operator between the crossed products that is equivariant with respect to the dual coactions. A similar result is proved for coactions and dual actions. It is shown that the only equivariant linear operators that lift twice through the action and dual coaction of an infinite group are the completely bounded ones.doi:10.2140/pjm.1989.139.155 fatcat:npltoizqczdwldzpuai6j7b5pa