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Local Proof of Algebraic Characterization of Free Actions

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. Here both tensor products are purely algebraic, and $\mathcal{O}(G)$ denotes the Hopf algebra of "polynomial" functions on $G$.