Compact Group Actions on Topological and Noncommutative Joins

Abstract

We consider the Type 1 and Type 2 noncommutative Borsuk-Ulam conjectures of Baum, D$ą$browski, and Hajac: there are no equivariant morphisms $A\to A{⊛}_{\delta }H$ or $H\to A{⊛}_{\delta }H$, respectively, when $H$ is a nontrivial compact quantum group acting freely on a unital $C^{\ast }$-algebra $A$. Here $A{⊛}_{\delta }H$ 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 $A{⊛}_{\delta }H$. 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.

15 pages. Version 4 has an expanded commentary in section 3, including the equivalence of a topological conjecture with a certain noncommutative generalization. To appear in Proceedings of the American Mathematical Society

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