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

Abstract

By analogy with the classical construction due to Forrest, Samei and Spronk, we associate to every compact quantum group [Formula: see text], a completely contractive Banach algebra [Formula: see text], which can be viewed as a deformed Fourier algebra of [Formula: see text]. To motivate the construction, we first analyze 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 Sołtan, the corresponding integration represented by a certain idempotent state on [Formula: see text] makes sense as long as [Formula: see text] is of Kac type. Finally, we analyze as an explicit example the algebras [Formula: see text], [Formula: see text], associated to Wang’s free orthogonal groups, and show that they are not operator weakly amenable.