Free Actions on C*-algebra Suspensions and Joins by Finite Cyclic Groups

TL;DR

This paper proves cases of the noncommutative Borsuk-Ulam conjecture for free actions of finite cyclic groups on unital C*-algebras, showing no equivariant maps exist to certain joins, and introduces new noncommutative join concepts.

Contribution

It provides a proof for specific cases of the noncommutative Borsuk-Ulam conjecture involving finite cyclic group actions on C*-algebras and proposes a novel noncommutative join construction.

Findings

No equivariant maps from A to its join with C(Z/kZ) under free actions.

Resolution of Dabrowski's conjecture for unreduced suspensions.

Introduction of a new noncommutative join concept leading to open problems.

Abstract

We present a proof for certain cases of the noncommutative Borsuk-Ulam conjectures proposed by Baum, D\k{a}browski, and Hajac. When a unital $C^*$-algebra $A$ admits a free action of $\mathbb{Z}/k\mathbb{Z}$, $k \geq 2$, there is no equivariant map from $A$ to the $C^*$-algebraic join of $A$ and the compact "quantum" group $C(\mathbb{Z}/k\mathbb{Z})$. This also resolves D\k{a}browski's conjecture on unreduced suspensions of $C^*$-algebras. Finally, we formulate a different type of noncommutative join than the previous authors, which leads to additional open problems for finite cyclic group actions.