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Equivariant Join and Fusion of Noncommutative Algebras
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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We translate the concept of the join of topological spaces to the language of $C^*$-algebras, replace the $C^*$-algebra of functions on the interval $[0,1]$ with evaluation maps at $0$ and $1$ by a unital $C^*$-algebra $C$ with appropriate two surjections, and introduce the notion of the fusion of unital $C^*$-algebras. An appropriate modification of this construction yields the fusion comodule algebra of a comodule algebra $P$ with the coacting Hopf algebra $H$. We prove that, if the comodule algebra $P$ is principal, then so is the fusion comodule algebra. When $C=C([0,1])$ and the two surjections are evaluation maps at $0$ and $1$, this result is a noncommutative-algebraic incarnation of the fact that, for a compact Hausdorff principal $G$-bundle $X$, the diagonal action of $G$ on the join $X*G$ is free.
Keywords: $C^*$-algebras; Hopf algebras; free actions. 
@article{SIGMA_2015_11_a81,
     author = {Ludwik D\k{a}browski and Tom Hadfield and Piotr M. Hajac},
     title = {Equivariant {Join} and {Fusion} of {Noncommutative} {Algebras}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2015},
     volume = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a81/}
}
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Ludwik Dąbrowski; Tom Hadfield; Piotr M. Hajac. Equivariant Join and Fusion of Noncommutative Algebras. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a81/

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