Realizations of free actions via their fixed point algebras

Schwieger, Kay ; Wagner, Stefan

Canadian mathematical bulletin, Tome 68 (2025) no. 2, pp. 568-581

Résumé

Let G be a compact group, let $\mathcal {B}$ be a unital C$^*$-algebra, and let $(\mathcal {A},G,\alpha )$ be a free C$^*$-dynamical system, in the sense of Ellwood, with fixed point algebra $\mathcal {B}$. We prove that $(\mathcal {A},G,\alpha )$ can be realized as the G-continuous part of the invariants of an equivariant coaction of G on a corner of $\mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}})$ for a certain Hilbert space ${\mathfrak {H}}$ that arises from the freeness of the action. This extends a result by Wassermann for free and ergodic C$^*$-dynamical systems. As an application, we show that any faithful $^*$-representation of $\mathcal {B}$ on a Hilbert space ${\mathfrak {H}}_{\mathcal {B}}$ gives rise to a faithful covariant representation of $(\mathcal {A},G,\alpha )$ on some truncation of ${\mathfrak {H}}_{\mathcal {B}} \otimes {\mathfrak {H}}$.

Détail

DOI : 10.4153/S0008439524000808

Mots-clés : Free action, coaction, fixed point algebra