Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map
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| Format: | Preprint |
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2021
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| author | De Commer, Kenny Martos, Rubén Nest, Ryszard |
| author_facet | De Commer, Kenny Martos, Rubén Nest, Ryszard |
| contents | We study the theory of projective representations for a compact quantum group $\mathbb{G}$, i.e. actions of $\mathbb{G}$ on $\mathcal{B}(H)$ for some Hilbert space $H$. We show that any such projective representation is inner, and is hence induced by an $Ω$-twisted representation for some unitary measurable $2$-cocycle $Ω$ on $\mathbb{G}$. We show that a projective representation is continuous, i.e. restricts to an action on the compact operators $\mathcal{K}(H)$, if and only if the associated $2$-cocycle is regular, and that this condition is automatically satisfied if $\mathbb{G}$ is of Kac type. This allows in particular to characterise the torsion of projective type of $\widehat{\mathbb{G}}$ in terms of the projective representation theory of $\mathbb{G}$. For a given regular unitary $2$-cocycle $Ω$, we then study $Ω$-twisted actions on C$^*$-algebras. We define deformed crossed products with respect to $Ω$, obtaining a twisted version of the Baaj-Skandalis duality and a quantum version of the Packer-Raeburn's trick. As an application, we provide a twisted version of the Green-Julg isomorphism and obtain the quantum Baum-Connes assembly map for permutation torsion-free discrete quantum groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_04365 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map De Commer, Kenny Martos, Rubén Nest, Ryszard Operator Algebras K-Theory and Homology Quantum Algebra Representation Theory We study the theory of projective representations for a compact quantum group $\mathbb{G}$, i.e. actions of $\mathbb{G}$ on $\mathcal{B}(H)$ for some Hilbert space $H$. We show that any such projective representation is inner, and is hence induced by an $Ω$-twisted representation for some unitary measurable $2$-cocycle $Ω$ on $\mathbb{G}$. We show that a projective representation is continuous, i.e. restricts to an action on the compact operators $\mathcal{K}(H)$, if and only if the associated $2$-cocycle is regular, and that this condition is automatically satisfied if $\mathbb{G}$ is of Kac type. This allows in particular to characterise the torsion of projective type of $\widehat{\mathbb{G}}$ in terms of the projective representation theory of $\mathbb{G}$. For a given regular unitary $2$-cocycle $Ω$, we then study $Ω$-twisted actions on C$^*$-algebras. We define deformed crossed products with respect to $Ω$, obtaining a twisted version of the Baaj-Skandalis duality and a quantum version of the Packer-Raeburn's trick. As an application, we provide a twisted version of the Green-Julg isomorphism and obtain the quantum Baum-Connes assembly map for permutation torsion-free discrete quantum groups. |
| title | Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map |
| topic | Operator Algebras K-Theory and Homology Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2112.04365 |