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 $\ast$-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. Following 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 a compact quantum group admits a representation whose \mbox{$K_1$-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.

References in corpus (4)

• Noncommutative Geometry Approach to Principal and Associated Bundles
• Free Actions on C*-algebra Suspensions and Joins by Finite Cyclic Groups
• Rank-two Milnor idempotents for the multipullback quantum complex projective plane
• The weak Hilbert-Smith conjecture from a Borsuk-Ulam-type conjecture

Cited by in corpus (3)

• Equivariant Dimensions of Graph C*-algebras
• The local-triviality dimension of actions of compact quantum groups
• Triviality of Equivariant Maps in Crossed Products and Matrix Algebras