Discrete Quantum Groups Research Articles

Uncertainty principles for locally compact quantum groups

In this paper, we prove the Donoho–Stark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the Hirschman–Beckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.

Operator Algebras in Rigid C*-Tensor Categories

In this article, we define operator algebras internal to a rigid C*-tensor category $\mathcal{C}$. A C*/W*-algebra object in $\mathcal{C}$ is an algebra object $\mathbf{A}$ in $\operatorname{ind}$-$\mathcal{C}$ whose category of free modules ${\sf FreeMod}_{\mathcal{C}}(\mathbf{A})$ is a $\mathcal{C}$-module C*/W*-category respectively. When $\mathcal{C}={\sf Hilb_{f.d.}}$, the category of finite dimensional Hilbert spaces, we recover the usual notions of operator algebras. We generalize basic representation theoretic results, such as the Gelfand-Naimark and von Neumann bicommutant theorems, along with the GNS construction. We define the notion of completely positive maps between C*-algebra objects in $\mathcal{C}$ and prove the analog of the Stinespring dilation theorem. As an application, we discuss approximation and rigidity properties, including amenability, the Haagerup property, and property (T) for a connected W*-algebra $\mathbf{M}$ in $\mathcal{C}$. Our definitions simultaneously unify the definitions of analytic properties for discrete quantum groups and rigid C*-tensor categories.

L2-Betti numbers of rigid C∗-tensor categories and discrete quantum groups

We compute the $L^2$-Betti numbers of the free $C^*$-tensor categories, which are the representation categories of the universal unitary quantum groups $A_u(F)$. We show that the $L^2$-Betti numbers of the dual of a compact quantum group $G$ are equal to the $L^2$-Betti numbers of the representation category $Rep(G)$ and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of $L^2$-Betti numbers for discrete quantum groups, including the quantum permutation groups and the free wreath product groups. Finally, we obtain upper bounds for the first $L^2$-Betti number in terms of a generating set of a $C^*$-tensor category.

Hopf images in locally compact quantum groups

This manuscript is devoted to the study of the concept of a generating subset (a.k.a. Hopf image of a morphism) in the setting of locally compact quantum groups. The aim of this paper is to provide an accurate description of the Hopf image of a given morphism. We extend and unify the previously existing approaches for compact and discrete quantum groups and present some results that can shed light on some local perspective in the theory of quantum groups. In particular, we provide a characterization of fullness of Hopf image in the language of partial actions as well as in representation-theoretic terms in the spirit of representation C⁎-categories, extending some known results to the broader setting of non-compact quantum groups.

Around Property (T) for Quantum Groups

We study Property (T) for locally compact quantum groups, providing several new characterisations, especially related to operator algebraic ergodic theory. Quantum Property (T) is described in terms of the existence of various Kazhdan type pairs, and some earlier structural results of Kyed, Chen and Ng are strengthened and generalised. For second countable discrete unimodular quantum groups with low duals, Property (T) is shown to be equivalent to Property (T)1,1 of Bekka and Valette. This is used to extend to this class of quantum groups classical theorems on ‘typical’ representations (due to Kerr and Pichot), and on connections of Property (T) with spectral gaps (due to Li and Ng) and with strong ergodicity of weakly mixing actions on a particular von Neumann algebra (due to Connes and Weiss). Finally, we discuss in the Appendix equivalent characterisations of the notion of a quantum group morphism with dense image.

Introduction to compact and discrete quantum groups

These are notes from introductory lectures at the graduate school "Topological Quantum Groups" in B\k{e}dlewo (June 28--July 11, 2015). The notes present the passage from Hopf algebras to compact quantum groups and sketch the notion of discrete quantum groups viewed as duals of compact quantum groups.

Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra

Banica and Vergnioux have shown that the dual discrete quantum group of a compact simply connected Lie group has polynomial growth of order the real manifold dimension. We extend this result to a general compact group and its topological dimension, by connecting it with the Gelfand-Kirillov dimension of an algebra. Furthermore, we show that polynomial growth for a compact quantum group G of Kac type implies *-regularity of the Fourier algebra A(G), that is every closed ideal of C(G) has a dense intersection with A(G). In particular, A(G) has a unique C*-norm.

Approximation properties for locally compact quantum groups

This is a survey of some aspects of the subject of approximation properties for locally compact quantum groups, based on lectures given at the {\it Topological Quantum Groups} Graduate School, 28 June - 11 July, 2015 in Bedlewo, Poland. We begin with a study of the dual notions of amenability and co-amenability, and then consider weakenings of these properties in the form of the Haagerup property and weak amenability. For discrete quantum groups, the interaction between these properties and various operator algebra approximation properties are investigated. We also study the connection between central approximation properties for discrete quantum groups and monoidal equivalence for their compact duals. We finish by discussing the central weak amenability and central Haagerup property for free quantum groups.

The spatial Rokhlin property for actions of compact quantum groups

We introduce the spatial Rokhlin property for actions of coexact compact quantum groups on C⁎-algebras, generalizing the Rokhlin property for both actions of classical compact groups and finite quantum groups. Two key ingredients in our approach are the concept of sequentially split ⁎-homomorphisms, and the use of braided tensor products instead of ordinary tensor products.We show that various structure results carry over from the classical theory to this more general setting. In particular, we show that a number of C⁎-algebraic properties relevant to the classification program pass from the underlying C⁎-algebra of a Rokhlin action to both the crossed product and the fixed point algebra. Towards establishing a classification theory, we show that Rokhlin actions exhibit a rigidity property with respect to approximate unitary equivalence. Regarding duality theory, we introduce the notion of spatial approximate representability for actions of discrete quantum groups. The spatial Rokhlin property for actions of a coexact compact quantum group is shown to be dual to spatial approximate representability for actions of its dual discrete quantum group, and vice versa.

Kirchberg's factorization property for discrete quantum groups

We show that the discrete duals of the universal unitary quantum groups and orthogonal quantum groups have Kirchberg's factorization property when n is different from 3.

Mean ergodic theorem for amenable discrete quantum groups and a Wiener-type theorem for compact metrizable groups

We prove a mean ergodic theorem for amenable discrete quantum groups. As an application, we prove a Wiener type theorem for continuous measures on compact metrizable groups.

Weak mixing for locally compact quantum groups

We generalize the notion of weakly mixing unitary representations to locally compact quantum groups, introducing suitable extensions of all standard characterizations of weak mixing to this setting. These results are used to complement the non-commutative Jacobs–de Leeuw–Glicksberg splitting theorem of Runde and the author [Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959]. Furthermore, a relation between mixing and weak mixing of state-preserving actions of discrete quantum groups and the properties of certain inclusions of von Neumann algebras, which is known for discrete groups, is demonstrated.

Compact quantum metric spaces from quantum groups of rapid decay

We present a modified version of the definition of property RD for discrete quantum groups given by Vergnioux in order to accommodate examples of non-unimodular quantum groups. Moreover we extend the construction of spectral triples associated to discrete groups with length functions, originally due to Connes, to the setting of quantum groups. For quantum groups of rapid decay we study the resulting spectral triples from the point of view of compact quantum metric spaces in the sense of Rieffel.

One-to-one correspondence between generating functionals and cocycles on quantum groups in presence of symmetry

We prove that under a symmetry assumption all cocycles on Hopf *-algebras arise from generating functionals. This extends earlier results of R.Vergnioux and D.Kyed and has two quantum group applications: all quantum L\'evy processes with symmetric generating functionals decompose into a maximal Gaussian and purely non-Gaussian part and the Haagerup property for discrete quantum groups is characterized by the existence of an arbitrary proper cocycle.

Lp{L_{p}}-representations of discrete quantum groups

Abstract Given a locally compact quantum group 𝔾 {\mathbb{G}} , we define and study representations and C ∗ {\mathrm{C}^{\ast}} -completions of the convolution algebra L 1 ⁢ ( 𝔾 ) {L_{1}(\mathbb{G})} associated with various linear subspaces of the multiplier algebra C b ⁢ ( 𝔾 ) {C_{b}(\mathbb{G})} . For discrete quantum groups 𝔾 {\mathbb{G}} , we investigate the left regular representation, amenability and the Haagerup property in this framework. When 𝔾 {\mathbb{G}} is unimodular and discrete, we study in detail the C ∗ {\mathrm{C}^{\ast}} -completions of L 1 ⁢ ( 𝔾 ) {L_{1}(\mathbb{G})} associated with the non-commutative L p {L_{p}} -spaces L p ⁢ ( 𝔾 ) {L_{p}(\mathbb{G})} . As an application of this theory, we characterize (for each p ∈ [ 1 , ∞ ) {p\in[1,\infty)} ) the positive definite functions on unimodular orthogonal and unitary free quantum groups 𝔾 {\mathbb{G}} that extend to states on the L p {L_{p}} - C ∗ {\mathrm{C}^{\ast}} -algebra of 𝔾 {\mathbb{G}} . Using this result, we construct uncountably many new examples of exotic quantum group norms for compact quantum groups.

Examples of factors which have no Cartan subalgebras

We consider some conditions similar to Ozawa’s condition (AO) and prove that if a non-injective factor satisfies such a condition and has the W ∗ C B A P \rm W^*CBAP , then it has no Cartan subalgebras. As a corollary, we prove that I I 1 \rm II_1 factors of universal orthogonal and unitary discrete quantum groups have no Cartan subalgebras. We also prove that continuous cores of type I I I 1 \rm III_1 factors with such a condition are semisolid as a I I ∞ \rm II_\infty factor.

On the structure of quantum automorphism groups

Abstract We compute the K-theory of quantum automorphism groups of finite-dimensional C * {C^{*}} -algebras in the sense of Wang. The results show in particular that the reduced C * {C^{*}} -algebras of functions on the quantum permutation groups S n + {S_{n}^{+}} are pairwise non-isomorphic for different values of n. Along the way we discuss some general facts regarding torsion in discrete quantum groups. In fact, the duals of quantum automorphism groups are the most basic examples of discrete quantum groups exhibiting genuine quantum torsion phenomena.

Property T for locally compact quantum groups

In this short paper, we obtained some equivalent formulations of property T for a general locally compact quantum group 𝔾, in terms of the full quantum group C*-algebras [Formula: see text] and the *-representation of [Formula: see text] associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if 𝔾 is of Kac type, we show that 𝔾 has property T if and only if every finite-dimensional irreducible *-representation of [Formula: see text] is an isolated point in the spectrum of [Formula: see text] (this also generalizes the corresponding locally compact group result). In addition, we give a way to construct property T discrete quantum groups using bicrossed products.

Residually finite quantum group algebras

We show that provided n≠3, the involutive Hopf ⁎-algebra Au(n) coacting universally on an n-dimensional Hilbert space has enough finite-dimensional representations in the sense that every non-zero element acts non-trivially in some finite-dimensional ⁎-representation. This implies that the discrete quantum group with group algebra Au(n) is maximal almost periodic (i.e. it embeds in its quantum Bohr compactification), answering a question posed by P. Sołtan in [21].We also prove analogous results for the involutive Hopf ⁎-algebra Bu(n) coacting universally on an n-dimensional Hilbert space equipped with a non-degenerate bilinear form.

Permanence of approximation properties for discrete quantum groups

We prove several results on the permanence of weak amenability and the Haagerup property for discrete quantum groups. In particular, we improve known facts on free products by allowing amalgamation over a finite quantum subgroup. We also define a notion of relative amenability for discrete quantum groups and link it with amenable equivalence of von Neumann algebras, giving additional permanence properties.