Noncommutative Borsuk-Ulam-type conjectures

Paul F. Baum, Ludwik Dąbrowski, and Piotr M. Hajac. "Noncommutative Borsuk-Ulam-type conjectures." Banach Center Publications 106.1 (2015): 9-18. <http://eudml.org/doc/281947>.

@article{PaulF2015,
abstract = {Within the framework of free actions of compact quantum groups on unital C*-algebras, we propose two conjectures. The first one states that, if $δ: A → A ⨂_\{min\} H$ is a free coaction of the C*-algebra H of a non-trivial compact quantum group on a unital C*-algebra A, then there is no H-equivariant *-homomorphism from A to the equivariant join C*-algebra $A ⊛_δ H$. For A being the C*-algebra of continuous functions on a sphere with the antipodal coaction of the C*-algebra of functions on ℤ/2ℤ, we recover the celebrated Borsuk-Ulam Theorem. The second conjecture states that there is no H-equivariant *-homomorphism from H to the equivariant join C*-algebra $A ⊛_δ H$. We show how to prove the conjecture in the special case $A = C(SU_q(2)) = H$, which is tantamount to showing the non-trivializability of Pflaum’s quantum instanton fibration built from $SU_q(2)$.},
author = {Paul F. Baum, Ludwik Dąbrowski, Piotr M. Hajac},
journal = {Banach Center Publications},
keywords = {noncommutative Borsuk-Ulam Theorem; noncommutative join; quantum sphere; quantum $SU\left( 2\right) $; quantum instanton bundle; compact quantum group; coaction},
language = {eng},
number = {1},
pages = {9-18},
title = {Noncommutative Borsuk-Ulam-type conjectures},
url = {http://eudml.org/doc/281947},
volume = {106},
year = {2015},
}

TY - JOUR
AU - Paul F. Baum
AU - Ludwik Dąbrowski
AU - Piotr M. Hajac
TI - Noncommutative Borsuk-Ulam-type conjectures
JO - Banach Center Publications
PY - 2015
VL - 106
IS - 1
SP - 9
EP - 18
AB - Within the framework of free actions of compact quantum groups on unital C*-algebras, we propose two conjectures. The first one states that, if $δ: A → A ⨂_{min} H$ is a free coaction of the C*-algebra H of a non-trivial compact quantum group on a unital C*-algebra A, then there is no H-equivariant *-homomorphism from A to the equivariant join C*-algebra $A ⊛_δ H$. For A being the C*-algebra of continuous functions on a sphere with the antipodal coaction of the C*-algebra of functions on ℤ/2ℤ, we recover the celebrated Borsuk-Ulam Theorem. The second conjecture states that there is no H-equivariant *-homomorphism from H to the equivariant join C*-algebra $A ⊛_δ H$. We show how to prove the conjecture in the special case $A = C(SU_q(2)) = H$, which is tantamount to showing the non-trivializability of Pflaum’s quantum instanton fibration built from $SU_q(2)$.
LA - eng
KW - noncommutative Borsuk-Ulam Theorem; noncommutative join; quantum sphere; quantum $SU\left( 2\right) $; quantum instanton bundle; compact quantum group; coaction
UR - http://eudml.org/doc/281947
ER -