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Realizations of free actions via their fixed point algebras
- Part of
-
- Journal:
- Canadian Mathematical Bulletin / Volume 68 / Issue 2 / June 2025
- Published online by Cambridge University Press:
- 17 January 2025, pp. 568-581
- Print publication:
- June 2025
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- Article
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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 
${\mathfrak {H}}_{\mathcal {B}} \otimes {\mathfrak {H}}$.
