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

Research output: Contribution to journal › Article › peer-review

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

Access to Document

10.3842/SIGMA.2014.060

Cite this

@article{a92544b4f1794df38c8128f9c9f065b5,

title = "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*-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.",

author = "Baum, \{Paul F.\} and Hajac, \{Piotr M.\}",

year = "2014",

month = jun,

day = "6",

doi = "10.3842/SIGMA.2014.060",

language = "English (US)",

volume = "10",

journal = "Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)",

issn = "1815-0659",

publisher = "Institute of Mathematics",

}

TY - JOUR

T1 - Local proof of algebraic characterization of free actions

AU - Baum, Paul F.

AU - Hajac, Piotr M.

PY - 2014/6/6

Y1 - 2014/6/6

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.

UR - https://www.scopus.com/pages/publications/84902081471

UR - https://www.scopus.com/pages/publications/84902081471#tab=citedBy

U2 - 10.3842/SIGMA.2014.060

DO - 10.3842/SIGMA.2014.060

M3 - Article

AN - SCOPUS:84902081471

SN - 1815-0659

VL - 10

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

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

M1 - 060

ER -

Local proof of algebraic characterization of free actions

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this