  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 $G$.
