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

SIGMA 11 (2015), 082, 7 pages      arXiv:1407.6020      https://doi.org/10.3842/SIGMA.2015.082

### Equivariant Join and Fusion of Noncommutative Algebras

Ludwik Dąbrowski a, Tom Hadfield b and Piotr M. Hajac c
a) SISSA (Scuola Internazionale Superiore di Studi Avanzati), Via Bonomea 265, 34136 Trieste, Italy
b) G-Research, Whittington House, 19-30 Alfred Place, London WC1E 7EA, UK
c) Institytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, 00-656 Warszawa, Poland

Received June 30, 2015, in final form October 03, 2015; Published online October 13, 2015

Abstract
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.

Key words: $C^*$-algebras; Hopf algebras; free actions.

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