Noncommutative Borsuk–Ulam-type conjectures revisited

Abstract

Let $H$ be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra $A$. It was recently conjectured that there does not exist an equivariant *-homomorphism from $A$ (type-I case) or $H$ (type-II case) to the equivariant noncommutative join C*-algebra $A{⊛}^{\delta }H$. When $A$ is the C*-algebra of functions on a sphere, and $H$ is the C*-algebra of functions on $\mathbb{Z}/2\mathbb{Z}$ acting antipodally on the sphere, then the conjecture of type I becomes the celebrated Borsuk–Ulam theorem. Taking advantage of recent work of Passer, we prove the conjecture of type I for compact quantum groups admitting a non-trivial torsion character. Next, we prove that, if the compact quantum group $(H,\Delta )$ admits a representation whose $K_1$-class is non-trivial and $A$ admits a character, then a stronger version of the type-II conjecture holds: the finitely generated projective module associated with $A{⊛}^{\delta }H$ via this representation is not stably free. In particular, we apply this result to the $q$-deformations of compact connected semisimple Lie groups and to the reduced group C*-algebras of free groups on $n>1$ generators.