Compact group actions on topological and noncommutative joins

Abstract:

We consider the Type 1 and Type 2 noncommutative Borsuk-Ulam conjectures of Baum, Dąbrowski, 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.