On the von Neumann algebra of class functions on a compact quantum group

TL;DR

This paper investigates the structure of class function algebras in compact quantum groups, revealing conditions under which they form maximal abelian subalgebras and exploring their properties in various quantum group settings.

Contribution

It introduces the concept of (quasi-)split inclusions for class function algebras and characterizes when these algebras are MASAs in different quantum group contexts.

Findings

Class function algebras are not MASAs in non-Kac free orthogonal quantum groups.

The inclusion of class function algebras can be quasi-split in certain unitary quantum groups.

Constructs bicrossed products where class functions form MASAs.

Abstract

We study analogues of the radial subalgebras in free group factors (called the algebras of class functions) in the setting of compact quantum groups. For the free orthogonal quantum groups we show that they are not MASAs, as soon as we are in a non-Kac situation. The most important notion to our present work is that of a (quasi-)split inclusion. We prove that the inclusion of the algebra of class functions is quasi-split for some unitary quantum groups; in this case the subalgebra is non-abelian and we also obtain a result concerning its relative commutant. In the positive direction, we construct certain bicrossed products from the quantum group $SU_q(2)$ for which the algebra of class functions is a MASA.