Compact Group Actions on Topological and Noncommutative Joins

TL;DR

This paper investigates noncommutative analogues of the Borsuk-Ulam conjecture involving compact quantum groups acting on C*-algebras, providing reductions, counterexamples, and conditions for validity.

Contribution

It introduces a modified noncommutative join construction and analyzes the Type 1 and Type 2 conjectures, offering new insights and examples in the context of quantum group actions.

Findings

Reduction of the Type 1 conjecture in the classical case

Counterexamples to the Type 2 conjecture

Conditions under which the Type 2 conjecture holds

Abstract

We consider the Type 1 and Type 2 noncommutative Borsuk-Ulam conjectures of Baum, D$\k{a}$browski, and Hajac: there are no equivariant morphisms $A \to A \circledast_\delta H$ or $H \to A \circledast_\delta H$, respectively, when $H$ is a nontrivial compact quantum group acting freely on a unital $C^*$-algebra $A$. Here $A \circledast_\delta H$ denotes the equivariant noncommutative join of $A$ and $H$; this join procedure is a modification of the topological join that allows a free action of $H$ on $A$ to produce a free action of $H$ on $A \circledast_\delta H$. For the classical case $H = \mathcal{C}(G)$, $G$ a compact group, we present a reduction of the Type 1 conjecture and counterexamples to the Type 2 conjecture. We also present some examples and conditions under which the Type 2 conjecture does hold.