Noncommutative Borsuk-Ulam-type conjectures revisited

TL;DR

This paper investigates noncommutative analogues of the Borsuk-Ulam theorem in the context of quantum groups and C*-algebras, proving conjectures for specific classes of quantum groups and exploring implications for noncommutative topology.

Contribution

It proves the type I Borsuk-Ulam conjecture for quantum groups with a non-trivial torsion character and establishes a stronger form of the type II conjecture under certain conditions.

Findings

Proved the type I conjecture for quantum groups with torsion characters.

Established that certain projective modules are not stably free under specified conditions.

Applied results to q-deformations of Lie groups and free group C*-algebras.

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. 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 $A\circledast^\delta H$ 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.