Remarks on the quantum Bohr compactification

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 * -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.

Section: New Examples Of Amenable Kac-type Quantum Groups With the Damentioning

confidence: 93%

On the similarity problem for locally compact quantum groups

Brannan

Youn

2019

“…Admissible finite-dimensional representations of locally compact quantum groups first appeared in the work of So ltan [39], who introduced the quantum Bohr compactification of a locally compact quantum group. Daws [15] studied further the quantum Bohr compactification as well as questions related to admissibility. It was conjectured (see [15,Conjecture 7.2]) that every finite-dimensional unitary representation of a locally compact quantum group is admissible.…”

Section: A Characterisation Of Admissible Finite-dimensional Unitary mentioning

confidence: 99%

“…The compact quantum group associated with AP(G) is the quantum Bohr compactification of G and is denoted by bG. See [39,15] for more details.…”

Section: A Characterisation Of Admissible Finite-dimensional Unitary mentioning

confidence: 99%

“…To enable this study, we prove the 'Admissibility Conjecture' for unimodular locally compact quantum groups, that is, we show that every finite-dimensional unitary representation of a unimodular locally compact quantum group is admissible. The Admissibility Conjecture is a long-standing open problem in the theory of quantum groups, which was implicitly stated in [39] and was conjectured in [15,Conjecture 7.2]. Admissibility of a finite-dimensional unitary representation of a quantum group means effectively that it 'factors' through a compact matrix quantum group.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Admissibility Conjecture and Kazhdan's Property (T) for quantum groups

Das

Daws

Salmi

2019

Self Cite

We give a partial solution to a long-standing open problem in the theory of quantum groups, namely we prove that all finite-dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups (Admissibility Conjecture for quantum group representations). We use this to study Kazhdan's Property (T) for quantum groups with non-trivial scaling group, strengthening and generalising some of the earlier results obtained by Fima, Kyed and So ltan, Chen and Ng, Daws, Skalski and Viselter, and Brannan and Kerr. Our main results are:(i) All finite-dimensional unitary representations of locally compact quantum groups which are either unimodular or arise through a special bicrossed product construction are admissible. (ii) A generalisation of a theorem of Wang which characterises Property (T) in terms of isolation of finite-dimensional irreducible representations in the spectrum. (iii) A very short proof of the fact that quantum groups with Property (T) are unimodular. (iv) A generalisation of a quantum version of a theorem of Bekka-Valette proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of non-existence of almost invariant vectors for weakly mixing representations. (v) A generalisation of a quantum version of Kerr-Pichot theorem, proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of denseness properties of weakly mixing representations.

“…For the Fourier algebra, this was suggested as early as [15]; recent work for quantum groups can be found in [25,Section 4] and [17], for example. However, outside of the commutative situation, to our knowledge there has been little study in terms of compactifications (for different notions of compactification, compare [8,27,28]). In particular, it is unknown, except in a few cases (see Section 7 below) if wap(L 1 (G)) is a C * -algebra.…”

Section: Introductionmentioning

confidence: 99%

Non-commutative separate continuity and weakly almost periodicity for Hopf von Neumann algebras

Daws

2015

Self Cite

For a compact Hausdorff space X, the space SC(X × X) of separately continuous complex valued functions on X can be viewed as a C * -subalgebra of C(X) * * ⊗C(X) * * , namely those elements which slice into C(X). The analogous definition for a non-commutative C * -algebra does not necessarily give an algebra, but we show that there is always a greatest C * -subalgebra. This thus gives a noncommutative notion of separate continuity. The tools involved are multiplier algebras and row/column spaces, familiar from the theory of Operator Spaces. We make some study of morphisms and inclusions. There is a tight connection between separate continuity and the theory of weakly almost periodic functions on (semi)groups. We use our non-commutative tools to show that the collection of weakly almost periodic elements of a Hopf von Neumann algebra, while itself perhaps not a C * -algebra, does always contain a greatest C * -subalgebra. This allows us to give a notion of non-commutative, or quantum, semitopological semigroup, and to briefly develop a compactification theory in this context.