Realizations of free actions via their fixed point algebras

Abstract

Let $G$ be a compact group, let $\mathcal{B}$ be a unital C$^*$-algebra, and let $(\mathcal{A},G,α)$ be a free C$^*$-dynamical system, in the sense of Ellwood, with fixed point algebra $\mathcal{B}$. We prove that $(\mathcal{A},G,α)$ 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$^*$-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 $(\mathcal{A},G,α)$ on some truncation of $\mathfrak{H}_{\mathcal{B}} \otimes \mathfrak{H}$.

Theorems & Definitions (28)

• Lemma 2.1
• proof
• Definition 2.2
• Lemma 2.3: cf.~SchWa20
• Remark 2.4
• Remark 2.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof