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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].

##### MSC:
 46L85 Noncommutative topology 58B32 Geometry of quantum groups 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory
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