Operator biflatness of the $L^1$-algebras of compact quantum groups

TL;DR

This paper demonstrates that the $L^1$-algebra of non-Kac type compact quantum groups lacks operator biflatness, providing insights into the structure and properties of quantum group algebras and their amenability.

Contribution

It establishes that non-Kac type compact quantum groups' $L^1$-algebras are not operator biflat, challenging existing conjectures about operator amenability and biprojectivity.

Findings

Non-Kac type compact quantum groups' $L^1$-algebras are not operator biflat.

Operator amenability implies operator biflatness, but the converse does not hold.

$L^1$-algebra of a locally compact quantum group is operator biprojective iff the group is compact and of Kac type.

Abstract

We prove that the $L^1$-algebra of any non-Kac type compact quantum group does not satisfy operator biflatness. Since operator amenability implies operator biflatness, this result shows that any co-amenable, non-Kac type compact quantum group gives a counter example to the conjecture that $L^1(\Gb)$ is operator amenable if and only if $\Gb$ is amenable and co-amenable for any locally compact quantum group $\Gb$. The result also implies that the $L^1$-algebra of a locally compact quantum group is operator biprojective if and only if $\Gb$ is compact and of Kac type.