$L_p$-Multipliers on Quantum Tori

TL;DR

This paper investigates bounded Fourier multipliers on noncommutative $L_p$-spaces of quantum tori, revealing differences from the completely bounded case by using transference techniques from the classical torus.

Contribution

It demonstrates that bounded multipliers on quantum tori depend on the parameter $ heta$, contrasting with the independence of completely bounded multipliers.

Findings

Bounded multipliers depend on the parameter $ heta$ of the quantum torus.

Transference methods from the classical torus are effective in analyzing these multipliers.

The results highlight a distinction between bounded and completely bounded multipliers on quantum tori.

Abstract

It was shown by Chen, Xu and Yin that completely bounded Fourier multipliers on noncommutative $L_p$-spaces of quantum tori $\mathbb T^d_\theta$ do not depend on the parameter $\theta$. We establish that the situation is somehow different for bounded multipliers. The arguments are based on transference from the commutative torus.