Compact Group Actions on Topological and Noncommutative Joins

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.