Realizations of free actions via their fixed point algebras

Abstract

Let $G$ be a compact group, let $\mathcal{B}$ be a unital C${}^{\ast }$-algebra, and let $(\mathcal{A},G,\alpha )$ be a free C${}^{\ast }$-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 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 C${}^{\ast }$-dynamical systems with trivial fixed point algebras. As an application, we show that any faithful \Star-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 ${\mathfrak{H}}_{\mathcal{B}}\otimes \mathfrak{H}$.

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