Twisted Fourier transforms on non-Kac compact quantum groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912889185501184 |
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| author | Youn, Sang-Gyun |
| author_facet | Youn, Sang-Gyun |
| contents | We introduce an analytic family of twisted Fourier transforms $\left\{\mathcal{F}^{(x)}_p\right\}_{x\in \mathbb{R},p\in [1,2)}$ for non-Kac compact quantum groups and establish a sharpened form of the Hausdorff-Young inequality in the range $0\leq x \leq 1$. Furthermore, we prove that the range $0\leq x \leq 1$ is both necessary and sufficient for the boundedness of $\mathcal{F}^{(x)}_p$ under the assumption of sub-exponential growth on the dual discrete quantum group. We also show that the range of boundedness of $\mathcal{F}^{(x)}_p$ can be strictly extended beyond $[0,1]$ for certain non-Kac and non-coamenable free orthogonal quantum groups. As applications, we derive a stronger form of the twisted rapid decay property for polynomially growing non-Kac discrete quantum groups, including the duals of the Drinfeld-Jimbo $q$-deformations, and construct an explicit contractive, but non-completely bounded, representation of the convolution algebra of any non-Kac free orthogonal quantum group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23316 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Twisted Fourier transforms on non-Kac compact quantum groups Youn, Sang-Gyun Operator Algebras Functional Analysis 43A15, 46L57, 46L52, 47A30 We introduce an analytic family of twisted Fourier transforms $\left\{\mathcal{F}^{(x)}_p\right\}_{x\in \mathbb{R},p\in [1,2)}$ for non-Kac compact quantum groups and establish a sharpened form of the Hausdorff-Young inequality in the range $0\leq x \leq 1$. Furthermore, we prove that the range $0\leq x \leq 1$ is both necessary and sufficient for the boundedness of $\mathcal{F}^{(x)}_p$ under the assumption of sub-exponential growth on the dual discrete quantum group. We also show that the range of boundedness of $\mathcal{F}^{(x)}_p$ can be strictly extended beyond $[0,1]$ for certain non-Kac and non-coamenable free orthogonal quantum groups. As applications, we derive a stronger form of the twisted rapid decay property for polynomially growing non-Kac discrete quantum groups, including the duals of the Drinfeld-Jimbo $q$-deformations, and construct an explicit contractive, but non-completely bounded, representation of the convolution algebra of any non-Kac free orthogonal quantum group. |
| title | Twisted Fourier transforms on non-Kac compact quantum groups |
| topic | Operator Algebras Functional Analysis 43A15, 46L57, 46L52, 47A30 |
| url | https://arxiv.org/abs/2503.23316 |