LpL_{p}-improving convolution operators on finite quantum groups

We characterize positive convolution operators on a finite quantum group $\mathbb{G}$ which are $L_{p}$-improving. More precisely, we prove that the convolution operator $T_{\varphi}:x\mapsto\varphi\star x$ given by a state $\varphi$ on $C(\mathbb{G})$ satisfies $$\exists1<p<2,\quad\|T_{\varphi}:L_{p}(\mathbb{G})\to L_{2}(\mathbb{G})\|=1$$ if and only if the Fourier series $\hat{\varphi}$ satisfy $\|\hat{\varphi}(\alpha)\|<1$ for all nontrivial irreducible unitary representations $\alpha$, if and only if the state $(\varphi\circ S)\star\varphi$ is non-degenerate (where $S$ is the antipode). We also prove that these $L_{p}$-improving properties are stable under taking free products, which gives a method to construct $L_{p}$-improving multipliers on infinite compact quantum groups. Our methods for non-degenerate states yield a general formula for computing idempotent states associated to Hopf images, which generalizes earlier work of Banica, Franz and Skalski.Comment: 20 pages. Final version. Minor correction

Pointwise convergence of noncommutative Fourier series

This paper is devoted to the study of pointwise convergence of Fourier series for compact groups, group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as some summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this work studies the refined counterpart of pointwise convergence of these Fourier series. We establish a general criterion of maximal inequalities for approximative identities of noncommutative Fourier multipliers. As a result we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to $1$ pointwise, so that the associated Fourier multipliers on noncommutative $L_p$-spaces satisfy the pointwise convergence for all $1<p<\infty$. In a similar fashion, we also obtain results for a large subclass of groups (as well as discrete quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fej\'{e}r means and Bochner-Riesz means in the noncommutative setting. Even back to the Fourier series of $L_p$-functions on Euclidean spaces and non-abelian compact groups, our results seem novel and yield new problems. On the other hand, we obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.Comment: v3: 83 pages; this version contains some corrections. v2: 74 pages; new results are added in Section 4, Section 5 and Section 6.