Equivariant Join and Fusion of Noncommutative Algebras

Abstract

We translate the concept of the join of topological spaces to the language of $C^{\ast }$-algebras, replace the $C^{\ast }$-algebra of functions on the interval $[0,1]$ with evaluation maps at $0$ and $1$ by a unital $C^{\ast }$-algebra $C$ with appropriate two surjections, and introduce the notion of the fusion of unital $C^{\ast }$-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 $X\ast G$ is free.

References in corpus (2)

• Free actions of compact quantum group on unital C*-algebras
• Piecewise principal comodule algebras

Cited by in corpus (10)

• Free actions of compact quantum group on unital C*-algebras
• Free Actions on C*-algebra Suspensions and Joins by Finite Cyclic Groups
• Towards a noncommutative Brouwer fixed-point theorem
• Fixed Point Algebras for Easy Quantum Groups
• Noncommutative Borsuk-Ulam-type conjectures revisited
• Equivariant Dimensions of Graph C*-algebras
• The local-triviality dimension of actions of compact quantum groups
• Invariants in Noncommutative Dynamics
• Triviality of Equivariant Maps in Crossed Products and Matrix Algebras
• Braided join comodule algebras of Galois objects