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

Abstract

We present a proof for certain cases of the noncommutative Borsuk-Ulam conjectures proposed by Baum, Dąbrowski, and Hajac. When a unital $C^{\ast }$-algebra $A$ admits a free action of $\mathbb{Z}/k\mathbb{Z}$, $k\ge 2$, there is no equivariant map from $A$ to the $C^{\ast }$-algebraic join of $A$ and the compact "quantum" group $C(\mathbb{Z}/k\mathbb{Z})$. This also resolves Dąbrowski's conjecture on unreduced suspensions of $C^{\ast }$-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.

13 pages. To appear in IUMJ. Version 4 restructures the results to address earlier work of Volovikov

References in corpus (2)

• Freeness of actions of finite groups on C*-algebras
• Towards a noncommutative Brouwer fixed-point theorem

Cited by in corpus (5)

• Noncommutative Borsuk-Ulam-type conjectures revisited
• Invariants in Noncommutative Dynamics
• The local-triviality dimension of actions of compact quantum groups
• Anticommutation in the Presentations of Theta-Deformed Spheres
• Triviality of Equivariant Maps in Crossed Products and Matrix Algebras