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

Abstract

We give some sufficient conditions for the injectivity of actions of compact quantum groups on C * -algebras. 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 * -algebras obtained by Rieffel deformations of compact, smooth manifolds.

Non-existence of genuine (compact) quantum symmetries of compact, connected smooth manifolds

“…that given any smooth action α of Q on C(M ) there is a Fréchet dense unital * -subalgebra C 0 of C ∞ (M ) on which α restricts to an algebraic co-action of Q 0 . It also follows (see [14], Corollary 3.3) that for any smooth action α, the corresponding reduced action α r is injective and hence it is implemented by some unitary representation.…”

Existence and rigidity of quantum isometry groups for compact metric spaces

“…It is proved in [15,Corollary 3.3]that for any smooth action α, the corresponding reduced action α r is injective and hence it is implemented by some unitary representation.…”