Local Proof of Algebraic Characterization of Free Actions

Abstract

Let $G$ be a compact Hausdorff topological group acting on a compact Hausdorff topological space $X$. Within the $C^{\ast }$-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 $G$.

References in corpus (3)

• Noncommutative Geometry Approach to Principal and Associated Bundles
• Free actions of compact quantum group on unital C*-algebras
• Piecewise principal comodule algebras

Cited by in corpus (2)

• Free actions of compact quantum group on unital C*-algebras
• Nontrivial Deformation of a Trivial Bundle