Harmonic operators on convolution quantum group algebras

TL;DR

This paper investigates harmonic operators on convolution algebras of locally compact quantum groups, characterizing their existence and linking them to properties like finiteness and amenability of the quantum group.

Contribution

It provides a characterization of non-zero harmonic operators in terms of quantum group properties, advancing understanding of harmonic analysis in quantum group settings.

Findings

Non-zero harmonic operators in compact quantum groups are characterized.

Existence of harmonic operators relates to quantum group properties such as amenability.

Results connect harmonic analysis with structural properties of quantum groups.

Abstract

Let ${\Bbb G}$ be a locally compact quantum group and ${\mathcal T}(L^2({\Bbb G}))$ be the Banach algebra of trace class operators on $L^2({\Bbb G})$ with the convolution induced by the right fundamental unitary of ${\Bbb G}$. We study the space of harmonic operators $\widetilde{\mathcal H}_\omega$ in ${\mathcal B}(L^2({\Bbb G}))$ associated to a contractive element $\omega\in {\mathcal T}(L^2({\Bbb G}))$. We characterize the existence of non-zero harmonic operators in ${\mathcal K}(L^2({\Bbb G}))$ and relate them with some properties of the quantum group ${\Bbb G}$, such as finiteness, amenability and co-amenability.