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\circledast^\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\circledast^\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.