Noncommutative Borsuk-Ulam-type conjectures revisited

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 circle dot(delta) 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 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 circle dot(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.