Equivariant Join and Fusion of Noncommutative Algebras

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.