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Partial actions of groups on C*-algebras and the closely related actions andcoactions of Hopf algebras received much attention over the last decades. Theyarise naturally as restrictions of their global counterparts to non-invariantsubalgebras, and the ambient eveloping global (co)actions have proven usefulfor the study of associated crossed products. In this article, we introduce thepartial coactions of C*-bialgebras, focussing on C*-quantum, and proveexistence of an enveloping global coaction under mild technical assumptions.The construction of the latter provides a left adjoint to the forgetful functorfrom coactions to partial coactions. We also show that partial coactions of thefunction algebra of a discrete group correspond to partial actions on directsummands of a C*-algebra, and relate partial coactions of a compact or its dualdiscrete C*-quantum group to partial coactions or partial actions of the denseHopf subalgebra.

Partial actions of C*-quantum groups I: Restriction and Globalization