On the similarity problem for locally compact quantum groups (Q2422467)

The Day-Dixmier theorem states that every bounded continuous representation of an amenable locally compact group on a Hilbert space is similar to a unitary representation. Due to the relation between bounded (respectively, unitary) continuous representations of \(G\) and bounded representations (respectively, \(^*\)-representations) of the group Banach algebra \(L^1(G)\) on Hilbert spaces, the Day-Dixmier theorem can be expressed in terms of certain similarity property of the involutive Banach algebra \(L^1(G)\). This paper answers, in the negative, the question of whether there is an analogue of the Day-Dixmier theorem in the case of locally compact quantum groups with coamenable dual quantum groups. More precisely, for a locally compact quantum group \(\mathbb{G}\), one denotes by \(L^1(\mathbb{G})\) the completely contractive Banach algebra of the predual of the von Neumann algebra \(L^\infty(\mathbb{G})\) representing \(\mathbb{G}\). There is a largest subspace (which is actually a subalgebra) \(L^1_\#(\mathbb{G})\subseteq L^1(\mathbb{G})\) on which an involution \(^\#\) compatible with the antipode of \(\mathbb{G}\) can be defined. A bounded representation of \(L^1(\mathbb{G})\) on a Hilbert space is called a \(^*\)-representation if its restriction on \(L^1_\#(\mathbb{G})\) is involutive. A locally compact quantum group \(\mathbb{G}\) is said to have the Day-Dixmier property if 1. every completely contractive representation of \(L^1(\mathbb{G})\) on a Hilbert space is a \(^*\)-representation; 2. every completely bounded representation of \(L^1(\mathbb{G})\) on a Hilbert space is similar to a completely contractive representation. The first main result of this paper is the following: Let \(\mathbb{G}\) be a compact quantum group such that the dimensions of its irreducible representations have subexponential growth. If \(\mathbb{G}\) has the Day-Dixmier property, then \(\mathbb{G}\) is of Kac-type. Using this, the authors give concrete classes of compact quantum groups that do not have the Day-Dixmier property. On the other hand, the authors give new classes of examples of Kac-type locally compact quantum groups that have the Day-Dixmier property. In particular, they show that every amenable discrete quantum group of Kac-type has the Day-Dixmier property.