Non-commutative ambits and equivariant compactifications

TL;DR

This paper establishes the existence of universal equivariant compactifications for actions of locally compact quantum groups on C*-algebras and explores their categorical properties, including limits, adjoints, and monomorphisms.

Contribution

It proves the existence of universal equivariant compactifications and analyzes their categorical structure, including local presentability and functor properties, for quantum group actions.

Findings

Universal equivariant compactification exists for quantum group actions.

Categories of compactifications are locally presentable, complete, and cocomplete.

Forgetful functors have colimit-creating adjoints and preserve limits under certain conditions.

Abstract

We prove that an action $\rho:A\to M(C_0(\mathbb{G})\otimes A)$ of a locally compact quantum group on a $C^*$-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 $\mathbb{G}$-$C^*$-algebras to unital $C^*$-algebras creates finite limits and is comonadic, and that the monomorphisms in the former category are injective.