On twisted Fourier analysis and convergence of Fourier series on discrete groups

Fourier series and twisted C*-crossed products

This paper is an invitation to Fourier analysis in the context of reduced twisted C*-crossed products associated with discrete unital twisted C*-dynamical systems. We discuss norm-convergence of Fourier series, multipliers and summation processes. Our study relies in an essential way on the (covariant and equivariant) representation theory of C*-dynamical systems on Hilbert C*-modules. It also yields some information on the ideal structure of reduced twisted C*-crossed products.

Twisted Fourier analysis and pseudo-probability distributions

We use a noncommutative generalization of Fourier analysis to define a broad class of pseudo-probability representations, which includes the known bosonic and discrete Wigner functions. We characterize the groups of quantum unitary operations which correspond to phase-space transformations, generalizing Gaussian and Clifford operations. As examples, we find Wigner representations for fermions, hard-core bosons, and angle-number systems.

扭曲 Fourier(-Stieltjes) 空间与顺从性

Twisted Fourier(-Stieltjes) spaces and amenability

本文通过群上的2-上闭链引入Fourier(-Stieltjes)代数的推广，研究相关乘子空间性质，并证明通过比较扭曲Fourier-Stieltjes空间与乘子空间刻画局部紧群顺从性的结论。

非Kac紧量子群上的扭傅里叶变换

Twisted Fourier transforms on non-Kac compact quantum groups

本文研究了非Kac紧量子群上的扭傅里叶变换，推导了Hausdorff-Young不等式的加强形式，并构造了非Kac自由正交量子群的显式收缩表示。

扭曲广群作用的傅里叶—斯蒂尔杰斯范畴

Fourier--Stieltjes category for twisted groupoid actions

该研究将傅里叶—斯蒂尔杰斯代数理论推广到étale广群对任意C*-丛的扭曲作用范畴，开发了相关工具并建立了KSGNS型膨胀结果，还提供了若干应用。

Twisted Hilbert transforms vs Kakeya sets of directions

Given a discrete group $\G$ and an orthogonal action $γ: \G \to O(n)$ we study $L_p$ convergence of Fourier integrals which are frequency supported on the semidirect product $\R^n \rtimes_γ\G$. Given a unit $u \in \R^n$ and $1 < p \neq 2 < \infty$, our main result shows that the twisted (directional) Hilbert transform $H_u \rtimes_γid_\G$ is $L_p$-bounded iff the orbit $\mathcal{O}_γ(u)$ is finite. This is in sharp contrast with twisted Riesz transforms $R_u \rtimes_γid_\G$, which are always bounded. Our result characterizes Fourier summability in $L_p$ for this class of groups. We also extend…