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

Abstract

Partial actions of groups on C*-algebras and the closely related actions and coactions of Hopf algebras received much attention over the last decades. They arise naturally as restrictions of their global counterparts to non-invariant subalgebras, and the ambient eveloping global (co)actions have proven useful for the study of associated crossed products. In this article, we introduce the partial coactions of C*-bialgebras, focussing on C*-quantum, and prove existence of an enveloping global coaction under mild technical assumptions. The construction of the latter provides a left adjoint to the forgetful functor from coactions to partial coactions. We also show that partial coactions of the function algebra of a discrete group correspond to partial actions on direct summands of a C*-algebra, and relate partial coactions of a compact or its dual discrete C*-quantum group to partial coactions or partial actions of the dense Hopf subalgebra.