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Local proof of algebraic characterization of free actions. (English) Zbl 1295.22010
Summary: 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 \(G\).

MSC:
22C05 Compact groups
55R10 Fiber bundles in algebraic topology
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
57S10 Compact groups of homeomorphisms
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