Partial actions of C∗-quantum groups

Abstract

Partial actions of groups on $C^{\ast }$-algebras and the closely related actions and coactions of Hopf algebras have received much attention in recent decades. They arise naturally as restrictions of their global counterparts to noninvariant subalgebras, and the ambient enveloping global (co)actions have proven useful for the study of associated crossed products. In this article, we introduce the partial coactions of $C^{\ast }$-bialgebras, focusing on $C^{\ast }$-quantum groups, and we prove the existence of an enveloping global coaction under mild technical assumptions. We also show that partial coactions of the function algebra of a discrete group correspond to partial actions on direct summands of a $C^{\ast }$-algebra, and we relate partial coactions of a compact or its dual discrete $C^{\ast }$-quantum group to partial coactions or partial actions of the dense Hopf subalgebra. As a fundamental example, we associate to every discrete $C^{\ast }$-quantum group a quantum Bernoulli shift.