Mathematics

Let $H$ be the $C^*$-algebra of a non-trivial compact quantum group acting freely on a unital $C^*$-algebra $A$. Baum, Dabrowski and Hajac conjectured that there does not exist an equivariant $*$-homomorphism from $A$ to the equivariant noncommutative join $C^*$-algebra $A^*H$. When $A$ is the $C^*$-algebra of functions on a sphere, and $H$ is the $C^*$-algebra of functions on $Z/2Z$  acting antipodally on the sphere, then the conjecture becomes the celebrated Borsuk-Ulam theorem. Recently, Passer proved the conjecture when $H$ is the commutative $C^*$-algebra of functions on a non-trivial compact group with a torsion element. The first goal of this talk is to show how to extend this result to the quantum setting. Next, with a stronger assumption that our compact quantum group is a $q$-deformation of a compact connected semisimple Lie group, we prove a stronger result that there exists a finite-dimensional representation of the compact quantum group such that, for any $C^*$-algebra $A$ admitting a character, the finitely generated projective module associated with $A*H$ via this representation is not stably free. (Based on joint work with L. Dabrowski and S. Neshveyev.)