# Noncommutative Borsuk-Ulam-type conjectures

Paul F. Baum; Ludwik Dąbrowski; Piotr M. Hajac

Banach Center Publications (2015)

- Volume: 106, Issue: 1, page 9-18
- ISSN: 0137-6934

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topPaul 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 -

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