Non-commutative ambits and equivariant compactifications

Abstract

We prove that an action $\rho :A\to M(C_0(\mathbb{G})\otimes A)$ of a locally compact quantum group on a $C^{\ast }$-algebra has a universal equivariant compactification and prove a number of other category-theoretic results on $\mathbb{G}$-equivariant compactifications: that the categories compactifications of $\rho$ and $A$, respectively, are locally presentable (hence complete and cocomplete), that the forgetful functor between them is a colimit-creating left adjoint, and that epimorphisms therein are surjective and injections are regular monomorphisms.

When $\mathbb{G}$ is regular, coamenable we also show that the forgetful functor from unital $\mathbb{G}$-$C^{\ast }$-algebras to unital $C^{\ast }$-algebras creates finite limits and is comonadic and that the monomorphisms in the former category are injective.