Equivariant Join and Fusion of Noncommutative Algebras

TL;DR

This paper develops a noncommutative algebraic analogue of the topological join and fusion constructions, proving that principality is preserved, which models free group actions on joins in a noncommutative setting.

Contribution

It introduces the fusion of unital C*-algebras and shows that principality is preserved under this construction, extending classical topological results to noncommutative algebras.

Findings

Fusion preserves principality of comodule algebras.

Noncommutative join models free group actions.

Generalizes classical topological join results.

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.