Noncommutative Borsuk-Ulam-type conjectures.

*(English)*Zbl 1343.46064
Ciccoli, Nicola (ed.) et al., From Poisson brackets to universal quantum symmetries. Selected papers of the workshop, IMPAN, Warsaw, August 18–22, 2014. Warsaw: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-29-4/pbk). Banach Center Publications 106, 9-18 (2015).

The authors first recall the classical Borsuk-Ulam theorem as the non-existence of a \(\mathbb{Z}_{2}\)-equivariant continuous map from \(\mathbb{S}^{n}\) to \(\mathbb{S}^{n-1}\), where the action of \(\mathbb{Z} _{2}\equiv\mathbb{Z}/\left( 2\mathbb{Z}\right) \) on the spheres is implemented by the antipodal map \(p\mapsto-p\). The classical join \(X\ast G\) of a topological \(G\)-space \(X\) with a compact group \(G\), defined as a quotient of \(\left[ 0,1\right] \times X\times G\), has a canonical diagonal \(G\)-action. It is pointed out that \(\mathbb{S}^{n-1}\) as a \(\mathbb{Z}_{2}\)-space is the same as the \(n\)-th iterated join \(\left( \mathbb{Z}_{2}\right) ^{\ast n}\) of the group \(\mathbb{Z}_{2}\) with itself, and the classical Borsuk-Ulam theorem leads to the conjecture that, for a compact Hausdorff group \(G\) acting freely on a compact Hausdorff space \(X\), there does not exist a \(G\)-equivariant continuous map from \(G\ast X\) to \(X\).

Via a change of variables \(\left( t,x,h\right) \mapsto\left( t,xh^{-1} ,h\right) \), the authors give a formulation of the \(G\)-space \(X\ast G\) such that the group \(G\) acts on its \(G\)-component only, which allows a direct generalization to a quantum version of join \(A\circledast_{\delta}H\) for a coaction \(\delta:A\rightarrow A\otimes_{\min}H\) of a compact quantum group \(H\) acting on a unital \(C^*\)-algebra \(A\). More precisely, by definition, the so-called equivariant noncommutative join \(A\circledast_{\delta}H\) of \(A\;\)and \(H\) consists of \(f\in C\left( \left[ 0,1\right] ,A\otimes_{\min}H\right) \) such that \(f\left( 0\right) \in\mathbb{C}\otimes H\) and \(f\left( 1\right) \in\delta\left( A\right) \), and it carries a canonical coaction \(\delta_{\Delta}\) of \(H\) which is free in the sense that the span of \(\left( x\otimes1\right) \delta_{\Delta}\left( y\right) \) with \(x,y\in A\circledast_{\delta}H\) is dense in \(\left( A\circledast_{\delta}H\right) \otimes_{\min}H\).

For a unital \(C^*\)-algebra \(A\) with a free coaction of a nontrivial compact quantum group \(H\), the authors propose a quantum version of the above conjecture as that there does not exist an \(H\)-equivariant *-homomorphism from \(A\) to \(A\circledast_{\delta}H\), and also propose a related conjecture that there does not exist an \(H\)-equivariant *-homomorphism from \(H\) to \(A\circledast_{\delta}H\). The latter is shown to hold for the case of the quantum group \(\mathrm{SU}_{q}\left( 2\right) \) acting on the \(C^*\)-algebra \(C\left( \mathrm{SU}_{q}\left( 2\right) \right) \) itself, concluding that Pflaum’s quantum instanton fibration \(C\left( \mathrm{SU}_{q}\left( 2\right) \right) \otimes_{\Delta}C\left( \mathrm{SU}_{q}\left( 2\right) \right) \) [M. J. Pflaum, Commun. Math. Phys. 166, No. 2, 279–315 (1994; Zbl 0824.58009)] is not trivializable.

For the entire collection see [Zbl 1332.81016].

Via a change of variables \(\left( t,x,h\right) \mapsto\left( t,xh^{-1} ,h\right) \), the authors give a formulation of the \(G\)-space \(X\ast G\) such that the group \(G\) acts on its \(G\)-component only, which allows a direct generalization to a quantum version of join \(A\circledast_{\delta}H\) for a coaction \(\delta:A\rightarrow A\otimes_{\min}H\) of a compact quantum group \(H\) acting on a unital \(C^*\)-algebra \(A\). More precisely, by definition, the so-called equivariant noncommutative join \(A\circledast_{\delta}H\) of \(A\;\)and \(H\) consists of \(f\in C\left( \left[ 0,1\right] ,A\otimes_{\min}H\right) \) such that \(f\left( 0\right) \in\mathbb{C}\otimes H\) and \(f\left( 1\right) \in\delta\left( A\right) \), and it carries a canonical coaction \(\delta_{\Delta}\) of \(H\) which is free in the sense that the span of \(\left( x\otimes1\right) \delta_{\Delta}\left( y\right) \) with \(x,y\in A\circledast_{\delta}H\) is dense in \(\left( A\circledast_{\delta}H\right) \otimes_{\min}H\).

For a unital \(C^*\)-algebra \(A\) with a free coaction of a nontrivial compact quantum group \(H\), the authors propose a quantum version of the above conjecture as that there does not exist an \(H\)-equivariant *-homomorphism from \(A\) to \(A\circledast_{\delta}H\), and also propose a related conjecture that there does not exist an \(H\)-equivariant *-homomorphism from \(H\) to \(A\circledast_{\delta}H\). The latter is shown to hold for the case of the quantum group \(\mathrm{SU}_{q}\left( 2\right) \) acting on the \(C^*\)-algebra \(C\left( \mathrm{SU}_{q}\left( 2\right) \right) \) itself, concluding that Pflaum’s quantum instanton fibration \(C\left( \mathrm{SU}_{q}\left( 2\right) \right) \otimes_{\Delta}C\left( \mathrm{SU}_{q}\left( 2\right) \right) \) [M. J. Pflaum, Commun. Math. Phys. 166, No. 2, 279–315 (1994; Zbl 0824.58009)] is not trivializable.

For the entire collection see [Zbl 1332.81016].

Reviewer: Albert Sheu (Lawrence)

##### MSC:

46L85 | Noncommutative topology |

58B32 | Geometry of quantum groups |

46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |

##### Keywords:

noncommutative Borsuk-Ulam theorem; noncommutative join; quantum sphere; quantum \(SU\left( 2\right) \); quantum instanton bundle; compact quantum group; coaction
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\textit{P. F. Baum} et al., Banach Cent. Publ. 106, 9--18 (2015; Zbl 1343.46064)

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