Noncommutative Borsuk-Ulam-type conjectures
Volume 106 / 2015
Abstract
Within the framework of free actions of compact quantum groups on unital C*-algebras, we propose two conjectures. The first one states that, if $\delta\colon A\to A\otimes_{\mathrm{min}}H$ is a free coaction of the C*-algebra $H$ of a non-trivial compact quantum group on a unital C*-algebra $A$, then there is no $H$-equivariant $*$-homomorphism from $A$ to the equivariant join C*-algebra $A\circledast_\delta H$. For $A$ being the C*-algebra of continuous functions on a sphere with the antipodal coaction of the C*-algebra of functions on $\mathbb{Z}/2\mathbb{Z}$, we recover the celebrated Borsuk–Ulam Theorem. The second conjecture states that there is no $H$-equivariant $*$-homomorphism from $H$ to the equivariant join C*-algebra $A\circledast_\delta H$. We show how to prove the conjecture in the special case $A=C(SU_q(2))=H$, which is tantamount to showing the non-trivializability of Pflaum’s quantum instanton fibration built from $SU_q(2)$.
