Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 060, 7 pages      arXiv:1402.3024      https://doi.org/10.3842/SIGMA.2014.060
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Local Proof of Algebraic Characterization of Free Actions

Paul F. Baum a, b and Piotr M. Hajac b, c
a) Mathematics Department, McAllister Building, The Pennsylvania State University, University Park, PA 16802, USA
b) Instytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, Warszawa, 00-656 Poland
c) Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ul. Hoża 74, 00-682 Warszawa, Poland

Received February 13, 2014, in final form May 21, 2014; Published online June 06, 2014

Abstract
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 $G$.

Key words: compact group; free action; Peter-Weyl-Galois condition.

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