A note on the injectivity of action by compact quantum groups on a class of $C^{\ast}$-algebras

TL;DR

This paper establishes conditions under which actions of compact quantum groups on $C^{}$-algebras are injective, with applications to smooth actions on manifolds and their deformations, enhancing understanding of quantum symmetries.

Contribution

It provides new sufficient conditions for the injectivity of quantum group actions on $C^{}$-algebras, including smooth manifolds and their Rieffel-deformations.

Findings

Faithful smooth actions on manifolds are injective.

Injectivity holds for actions on Rieffel-deformed $C^{}$-algebras.

Conditions for injectivity are explicitly characterized.

Abstract

We give some sufficient conditions for the injectivity of actions of compact quantum groups on $C^{\ast}$-algebra. As an application, we prove that any faithful smooth action by a compact quantum group on a compact smooth (not necessarily connected) manifold is injective. A similar result is proved for actions on $C^{\ast}$- algebras obtained by Rieffel-deformation of compact, smooth manifolds.