# Noncommutative Borsuk–Ulam-type conjectures revisited

### Ludwik Dąbrowski

SISSA, Trieste, Italy### Piotr M. Hajac

IMPAN, Warsaw, Poland### Sergey Neshveyev

University of Oslo, Norway

## 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 $A⊛_{δ}H$. 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 $(H,Δ)$ 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 $A⊛_{δ}H$ 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.

## Cite this article

Ludwik Dąbrowski, Piotr M. Hajac, Sergey Neshveyev, Noncommutative Borsuk–Ulam-type conjectures revisited. J. Noncommut. Geom. 14 (2020), no. 4, pp. 1305–1324

DOI 10.4171/JNCG/352