Integration over the quantum diagonal subgroup and associated Fourier-like algebras

TL;DR

This paper constructs a quantum analog of Fourier algebras for compact quantum groups, analyzing the quantum diagonal subgroup concept and examining properties of associated algebras, including non-amenability in specific cases.

Contribution

It introduces a new Fourier-like algebra for compact quantum groups based on quantum diagonal subgroup integration, despite the non-existence of actual quantum diagonal subgroups.

Findings

The algebra $A_\Delta(\mathbb{G})$ is a completely contractive Banach algebra.

Quantum diagonal subgroups do not exist, but their associated integration makes sense for Kac type groups.

The algebra $A_\Delta(O_N^+)$ is not operator weakly amenable.

Abstract

By analogy with the classical construction due to Forrest, Samei and Spronk we associate to every compact quantum group $\mathbb{G}$ a completely contractive Banach algebra $A_\Delta(\mathbb{G})$, which can be viewed as a deformed Fourier algebra of $\mathbb{G}$. To motivate the construction we first analyse in detail the quantum version of the integration over the diagonal subgroup, showing that although the quantum diagonal subgroups in fact never exist, as noted earlier by Kasprzak and So{\l}tan, the corresponding integration represented by a certain idempotent state on $C(\mathbb{G})$ makes sense as long as $\mathbb{G}$ is of Kac type. Finally we analyse as an explicit example the algebras $A_\Delta(O_N^+)$, $N\ge 2$, associated to Wang's free orthogonal groups, and show that they are not operator weakly amenable.