# zbMATH — the first resource for mathematics

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
##### Keywords:
compact group; free action; Peter-Weyl-Galois condition
Full Text: