On the similarity problem for locally compact quantum groups

TL;DR

This paper investigates the analogue of the Day-Dixmier theorem for locally compact quantum groups, demonstrating its failure for non-Kac types and confirming its validity for certain Kac-type quantum groups.

Contribution

It proves the conjecture fails for many non-Kac compact quantum groups but holds for several Kac-type quantum groups, expanding understanding of representation theory in quantum groups.

Findings

The conjecture is false for non-Kac type quantum groups, including q-deformations.

The theorem holds for certain Kac-type quantum groups, such as amenable discrete quantum groups.

Provides new classes of quantum groups where the Day-Dixmier theorem applies.

Abstract

A well-known theorem of Day and Dixmier states that any uniformly bounded representation of an amenable locally compact group $G$ on a Hilbert space is similar to a unitary representation. Within the category of locally compact quantum groups, the conjectural analogue of the Day-Dixmier theorem is that every completely bounded Hilbert space representation of the convolution algebra of an amenable locally compact quantum group should be similar to a $\ast$-representation. We prove that this conjecture is false for a large class of non-Kac type compact quantum groups, including all $q$-deformations of compact simply connected semisimple Lie groups. On the other hand, within the Kac framework, we prove that the Day-Dixmier theorem does indeed hold for several new classes of examples, including amenable discrete quantum groups of Kac-type.