A Note on Operator Biprojectivity of Compact Quantum Groups

TL;DR

This paper investigates the conditions under which the convolution algebra of a locally compact quantum group is operator biprojective, establishing links to compactness and Kac algebra properties, with implications for the structure of quantum groups.

Contribution

It shows that if the splitting morphism for operator biprojectivity is completely positive or contractive, then the quantum group must be a Kac algebra, providing new insights into the structure of quantum groups.

Findings

Operator biprojectivity implies compactness for quantum groups.

Complete positivity of the splitting morphism characterizes Kac algebras.

Modular properties of the Haar state are crucial in the analysis.

Abstract

Given a (reduced) locally compact quantum group $A$, we can consider the convolution algebra $L^1(A)$ (which can be identified as the predual of the von Neumann algebra form of $A$). It is conjectured that $L^1(A)$ is operator biprojective if and only if $A$ is compact. The "only if" part always holds, and the "if" part holds for Kac algebras. We show that if the splitting morphism associated with $L^1(A)$ being biprojective can be chosen to be completely positive, or just contractive, then we already have a Kac algebra. We give another proof of the converse, indicating how modular properties of the Haar state seem to be important.