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Within the $C^{*}$-algebra $C(X)$ of all continuous complex-valued functions on $X$, there is the Peter-Weyl algebra $\\mathcal{P}_G(X)$ which is the (purely algebraic) direct sum of the isotypical components for the action of $G$ on $C(X)$. We prove that the action of $G$ on $X$ is free if and only if the canonical map $\\mathcal{P}_G(X)\\otimes_{C(X/G)}\\mathcal{P}_G(X)\\to \\mathcal{P}_G(X)\\otimes\\mathcal{O}(G)$ is bijective. 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Baum, Piotr M. Hajac","submitted_at":"2014-02-13T03:19:21Z","abstract_excerpt":"Let $G$ be a compact Hausdorff topological group acting on a compact Hausdorff topological space $X$. Within the $C^{*}$-algebra $C(X)$ of all continuous complex-valued functions on $X$, there is the Peter-Weyl algebra $\\mathcal{P}_G(X)$ which is the (purely algebraic) direct sum of the isotypical components for the action of $G$ on $C(X)$. We prove that the action of $G$ on $X$ is free if and only if the canonical map $\\mathcal{P}_G(X)\\otimes_{C(X/G)}\\mathcal{P}_G(X)\\to \\mathcal{P}_G(X)\\otimes\\mathcal{O}(G)$ is bijective. 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