On Harmonic Noncommutative $L^p$-Operators on Locally Compact Quantum Groups

TL;DR

This paper investigates harmonic operators in non-commutative L^p-spaces associated with locally compact quantum groups, showing that non-degenerate quantum measures lead to trivial harmonic operator spaces across all p.

Contribution

It establishes the triviality of harmonic operators in non-commutative L^p-spaces for non-degenerate quantum measures on locally compact quantum groups.

Findings

H_mu^p is trivial for all 1 ≤ p < ∞ when μ is non-degenerate.

The result extends classical harmonic analysis to the quantum group setting.

Provides a characterization of harmonic operators in non-commutative L^p-spaces.

Abstract

For a locally compact quantum group $\G$ with tracial Haar weight $\varphi$, and a quantum measure $\mu$ on $\G$, we study the space ${H}_\mu^p$ of $\mu$-harmonic operators in the non-commutative $L^p$-space ${L}^p(\G)$ associated to the Haar weight $\varphi$. The main result states that if $\mu$ is non-degenerate, then ${H}_\mu^p$ is trivial for all $1 \leq p < \infty$.