Local proof of algebraic characterization of free actions

Paul F. Baum; Piotr M. Hajac

doi:10.3842/SIGMA.2014.060

Local proof of algebraic characterization of free actions

Paul F. Baum, Piotr M. Hajac

Mathematics

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3 Scopus citations

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 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 P_G(X)⊗C(X/G)P_G(X)→P_G(X)⊗O(G) is bijective. Here both tensor products are purely algebraic, and O(G) denotes the Hopf algebra of "polynomial" functions on G.

Original language	English (US)
Article number	060
Journal	Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume	10
DOIs	https://doi.org/10.3842/SIGMA.2014.060
State	Published - Jun 6 2014

All Science Journal Classification (ASJC) codes

Analysis
Mathematical Physics
Geometry and Topology

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10.3842/SIGMA.2014.060

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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 PG(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 PG(X)⊗C(X/G)PG(X)→PG(X)⊗O(G) is bijective. Here both tensor products are purely algebraic, and O(G) denotes the Hopf algebra of {"}polynomial{"} functions on G.",

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N2 - 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 PG(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 PG(X)⊗C(X/G)PG(X)→PG(X)⊗O(G) is bijective. Here both tensor products are purely algebraic, and O(G) denotes the Hopf algebra of "polynomial" functions on G.

AB - 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 PG(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 PG(X)⊗C(X/G)PG(X)→PG(X)⊗O(G) is bijective. Here both tensor products are purely algebraic, and O(G) denotes the Hopf algebra of "polynomial" functions on G.

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Local proof of algebraic characterization of free actions

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