Compact Quantum Groups Research Articles

SAT actions of discrete quantum groups and minimal injective extensions of their von Neumann algebras

We introduce a natural generalization of the notion of strongly approximately transitive (SAT) states for actions of locally compact quantum groups. In the case of discrete quantum groups of Kac type, we show that the existence of unique stationary SAT states entails rigidity results concerning injective extensions of quantum group von Neumann algebras.

Braided quantum symmetries of graph C∗-algebras

We prove the existence of a universal braided compact quantum group acting on a graph [Formula: see text]-algebra in the category of [Formula: see text]-[Formula: see text]-algebras with a twisted monoidal structure, in the spirit of the seminal work of Wang. To achieve this, we construct a braided analogue of the free unitary quantum group and study its bosonization. As a concrete example, we compute this universal braided compact quantum group for the Cuntz algebra.

Rigidity on quantum symmetry for a certain class of graph C*-algebras

Quantum symmetry of graph C*-algebras has been studied, under the consideration of different formulations, in the past few years. It is already known that the compact quantum group (C(S1)∗C(S1)∗⋯∗C(S1)︸|E(Γ)|−times,Δ) [in short, *|E(Γ)|C(S1),Δ] always acts on a graph C*-algebra for a finite, connected, directed graph Γ in the category introduced by Joardar and Mandal, where |E(Γ)| ≔ number of edges in Γ. In this article, we show that for a certain class of graphs including Toeplitz algebra, quantum odd sphere, matrix algebra etc. the quantum symmetry of their associated graph C*-algebras remains *|E(Γ)|C(S1),Δ in the category as mentioned before. More precisely, if a finite, connected, directed graph Γ satisfies the following graph theoretic properties: (i) there does not exist any cycle of length ≥2 (ii) there exists a path of length (|V(Γ)| − 1) which consists all the vertices, where |V(Γ)| ≔ number of vertices in Γ (iii) given any two vertices (may not be distinct) there exists at most one edge joining them, then the universal object coincides with *|E(Γ)|C(S1),Δ. Furthermore, we have pointed out a few counter examples whenever the above assumptions are violated.

Topological boundaries of representations and coideals

For a locally compact quantum group G, a (left) coideal is a (left) G-invariant von Neumann subalgebra of L∞(G). We introduce and analyze various generalizations of amenability and coamenability to coideals of discrete and compact quantum groups. We focus on a particular class of coideals found in the category of compact quantum groups, which are associated with a compact quasi-subgroup. This class includes all coideals of the quotient type. We also introduce the notion of a Furstenberg-Hamana boundary for representations of discrete quantum groups and use it to study amenability and coamenability properties of coideals. We then prove that a coideal of a compact quantum group that is associated with a compact quasi-subgroup is coamenable if and only if its codual coideal is G-injective. If G is a unimodular or an exact discrete quantum group, we can replace G-injectivity in the latter statement with the weaker condition of relative amenability. This result leads to a complete characterization of the unique trace property. Specifically, a unimodular discrete quantum group G has the unique trace property if and only if the action of G on its noncommutative Furstenberg boundary is faithful. We also demonstrated that if a unimodular discrete quantum group G is C⁎-simple then it has the unique trace property. These findings are the quantum analogs of the groundbreaking results of Breuillard, Kalantar, Kennedy, and Ozawa and they provide answers to questions posted by Kalantar, Kasprzak, Skalski, and Vergnioux.

Actions of Compact and Discrete Quantum Groups on Operator Systems

Abstract We introduce the notion of an action of a discrete or compact quantum group on an operator system, and study equivariant operator system injectivity. We then prove a duality result that relates equivariant injectivity with dual injectivity of associated crossed products. As an application, we give a description of the equivariant injective envelope of the reduced crossed product built from an action of a discrete quantum group on an operator system.

Convolution semigroups on Rieffel deformations of locally compact quantum groups

Convolution semigroups on Rieffel deformations of locally compact quantum groups

Quantization of locally compact groups associated with essentially bijective 1-cocycles

Given an extension [Formula: see text] of locally compact groups, with [Formula: see text] abelian, and a compatible essentially bijective [Formula: see text]-cocycle [Formula: see text], we define a dual unitary [Formula: see text]-cocycle on [Formula: see text] and show that the associated deformation of [Formula: see text] is a cocycle bicrossed product defined by a matched pair of subgroups of [Formula: see text]. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang–Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of [Formula: see text] on [Formula: see text] and a unitary quantization map [Formula: see text] of Kohn–Nirenberg type.

Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras

AbstractWe find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with one‐dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter–Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, Härtel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then, we make use of the fact that the free unitary quantum groups and some of its non‐Kac versions can be realized as a glued free product of a (non‐Kac) free orthogonal quantum group with , the finite group of order 2. To obtain the resolution also for more general universal cosovereign Hopf algebras, we extend Gromada's proof from compact quantum groups to the framework of matrix Hopf algebras. As a by‐product of this approach, we also obtain a projective resolution for the freely modified bistochastic quantum groups. Only a special subclass of free unitary quantum groups and universal cosovereign Hopf algebras decompose as a glued free product in the described way. In order to verify that the sequence we found is a free resolution in general (as long as the parameter matrix is generic, two conditions which are automatically fulfilled in the free unitary quantum group case), we use the theory of Hopf bi‐Galois objects and Bichon's results on monoidal equivalences between the categories of Yetter–Drinfeld modules over universal cosovereign Hopf algebras for different parameter matrices.

Entanglement-symmetries of covariant channels

Let G and G′ be monoidally equivalent compact quantum groups, and let H be a Hopf-Galois object realising a monoidal equivalence between these groups' representation categories. This monoidal equivalence induces an equivalence Chan(G) → Chan(G′), where Chan(G) is the category whose objects are finite-dimensional C∗-algebras with an action of G and whose morphisms are covariant channels. We show that, if the Hopf-Galois object H has a finite-dimensional *-representation, then channels related by this equivalence can simulate each other using a finite-dimensional entangled resource. We use this result to calculate the entanglement-assisted capacities of certain quantum channels.

Quantum graphs: Different perspectives, homomorphisms and quantum automorphisms

We undertake a study of the notion of a quantum graph over arbitrary finite-dimensional C ∗ C^* -algebras B B equipped with arbitrary faithful states. Quantum graphs are realised principally as either certain operators on L 2 ( B ) L^2(B) , the quantum adjacency matrices, or as certain operator bimodules over B ′ B’ . We present a simple, purely algebraic approach to proving equivalence between these settings, thus recovering existing results in the tracial state setting. For non-tracial states, our approach naturally suggests a generalisation of the operator bimodule definition, which takes account of (some aspect of) the modular automorphism group of the state. Furthermore, we show that each such “non-tracial” quantum graph corresponds to a “tracial” quantum graph which satisfies an extra symmetry condition. We study homomorphisms (or CP-morphisms) of quantum graphs arising from unital completely positive (UCP) maps, and the closely related examples of quantum graphs constructed from UCP maps. We show that these constructions satisfy automatic bimodule properties. We study quantum automorphisms of quantum graphs, give a definition of what it means for a compact quantum group to act on an operator bimodule, and prove an equivalence between this definition, and the usual notion defined using a quantum adjacency matrix. We strive to give a relatively self-contained, elementary, account, in the hope this will be of use to other researchers in the field.

Cocycle Twisting of Semidirect Products and Transmutation

Abstract We apply Majid’s transmutation procedure to Hopf algebra maps $H \to {{\mathbb {C}}}[T]$, where $T$ is a compact abelian group, and explain how this construction gives rise to braided Hopf algebras over quotients of $T$ by subgroups that are cocentral in $H$. This allows us to unify and generalize a number of recent constructions of braided compact quantum groups, starting from the braided $SU_{q}(2)$ quantum group, and describe their bosonizations.

Examples of compact quantum groups with [formula omitted] a factor

For each λ∈]0,1] we exhibit an uncountable family of compact quantum groups G such that the von Neumann algebra L∞(G) is the injective factor of type IIIλ with separable predual. We also show that uncountably many injective factors of type III0 arise as L∞(G) for some compact quantum group G. We introduce natural invariants of quantum groups related to the scaling group, modeled on the Connes invariant T for von Neumann algebras. We compute the values of these invariants in our examples, which allows us to distinguish between them. We also investigate their connection to the Connes invariants T(L∞(G)), S(L∞(G)).In the final section we show that for a compact quantum group G the von Neumann algebra L∞(G) cannot have a direct summand of the form B(H) with dim⁡H=∞. In particular, factors of type I (of any dimension) cannot be obtained as L∞(G) for a non-trivial compact quantum group G.

The approximation property for locally compact quantum groups

We study the Haagerup–Kraus approximation property for locally compact quantum groups, generalising and unifying previous work by Kraus–Ruan and Crann. Along the way we discuss how multipliers of quantum groups interact with the C⁎-algebraic theory of locally compact quantum groups. Several inheritance properties of the approximation property are established in this setting, including passage to quantum subgroups, free products of discrete quantum groups, and duals of double crossed products. We also discuss a relation to the weak⁎ operator approximation property. For discrete quantum groups, we introduce a central variant of the approximation property, and relate this to a version of the approximation property for rigid C⁎-tensor categories, building on work of Arano–De Laat–Wahl.

Equivariant -correspondences and compact quantum group actions on Pimsner algebras

Abstract Let G be a compact quantum group. We show that given a G-equivariant $\textrm {C}^*$ -correspondence E, the Pimsner algebra $\mathcal {O}_E$ can be naturally made into a G- $\textrm {C}^*$ -algebra. We also provide sufficient conditions under which it is guaranteed that a G-action on the Pimsner algebra $\mathcal {O}_E$ arises in this way, in a suitable precise sense. When G is of Kac type, a KMS state on the Pimsner algebra, arising from a quasi-free dynamics, is G-equivariant if and only if the tracial state obtained from restricting it to the coefficient algebra is G-equivariant, under a natural condition. We apply these results to the situation when the $\textrm {C}^*$ -correspondence is obtained from a finite, directed graph and draw various conclusions on the quantum automorphism groups of such graphs, both in the sense of Banica and Bichon.

Non-commutative ambits and equivariant compactifications

We prove that an action \rho:A\to M(C_0(\mathbb{G})\otimes A) of a locally compact quantum group on a C^* -algebra has a universal equivariant compactification and prove a number of other category-theoretic results on \mathbb{G} -equivariant compactifications: that the categories compactifications of \rho and A , respectively, are locally presentable (hence complete and cocomplete), that the forgetful functor between them is a colimit-creating left adjoint, and that epimorphisms therein are surjective and injections are regular monomorphisms. When \mathbb{G} is regular, coamenable we also show that the forgetful functor from unital \mathbb{G} - C^* -algebras to unital C^* -algebras creates finite limits and is comonadic and that the monomorphisms in the former category are injective.

Crystal limits of compact semisimple quantum groups as higher-rank graph algebras

Abstract Let O q ⁢ [ K ] \mathcal{O}_{q}[K] be the quantized coordinate ring over the field C ⁢ ( q ) \mathbb{C}(q) of rational functions corresponding to a compact semisimple Lie group 𝐾, equipped with its ∗-structure. Let A 0 ⊂ C ⁢ ( q ) {\mathbf{A}_{0}}\subset\mathbb{C}(q) denote the subring of regular functions at q = 0 q=0 . We introduce an A 0 \mathbf{A}_{0} -subalgebra O q A 0 ⁢ [ K ] ⊂ O q ⁢ [ K ] \mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K]\subset\mathcal{O}_{q}[K] which is stable with respect to the ∗-structure and which has the following properties with respect to the crystal limit q → 0 q\to 0 . The specialization of O q ⁢ [ K ] \mathcal{O}_{q}[K] at each q ∈ ( 0 , ∞ ) ∖ { 1 } q\in(0,\infty)\setminus\{1\} admits a faithful ∗-representation π q \pi_{q} on a fixed Hilbert space, a result due to Soibelman. We show that, for every element a ∈ O q A 0 ⁢ [ K ] a\in\mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K] , the family of operators π q ⁢ ( a ) \pi_{q}(a) admits a norm limit as q → 0 q\to 0 . These limits define a ∗-representation π 0 \pi_{0} of O q A 0 ⁢ [ K ] \mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K] . We show that the resulting ∗-algebra O ⁢ [ K 0 ] = π 0 ⁢ ( O q A 0 ⁢ [ K ] ) \mathcal{O}[K_{0}]=\pi_{0}(\mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K]) is a Kumjian–Pask algebra, in the sense of Aranda Pino, Clark, an Huef and Raeburn. We give an explicit description of the underlying higher-rank graph in terms of crystal basis theory. As a consequence, we obtain a continuous field of C * C^{*} -algebras ( C ⁢ ( K q ) ) q ∈ [ 0 , ∞ ] (C(K_{q}))_{q\in[0,\infty]} , where the fibres at q = 0 q=0 and ∞ are explicitly defined higher-rank graph algebras.

Invariant means and multipliers on convolution quantum group algebras

Let [Formula: see text] be a locally compact quantum group. Then the space [Formula: see text] of trace class operators on [Formula: see text] is a Banach algebra with the convolution induced by the right fundamental unitary of [Formula: see text]. We show that properties of [Formula: see text] such as amenability, triviality and compactness are equivalent to the existence of left or right invariant means on the convolution Banach algebra [Formula: see text]. We also investigate the relation between the existence of certain (weakly) compact right and left multipliers of [Formula: see text] and some properties of [Formula: see text].

Compact quantum group structures on type-I $\mathrm{C}^*$-algebras

We prove a number of results having to do with equipping type-I \mathrm{C}^* -algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the \mathrm{C}^* -algebra in question is an extension of a non-zero finite direct sum of elementary \mathrm{C}^* -algebras by a commutative unital \mathrm{C}^* -algebra then it must be finite-dimensional.

L_p–L_q Fourier Multipliers on Locally Compact Quantum Groups

Let mathbb {G} be a locally compact quantum group with dual widehat{mathbb {G}}. Suppose that the left Haar weight varphi and the dual left Haar weight widehat{varphi } are tracial, e.g. mathbb {G} is a unimodular Kac algebra. We prove that for 1<ple 2 le q<infty , the Fourier multiplier m_{x} is bounded from L_p(widehat{mathbb {G}},widehat{varphi }) to L_q(widehat{mathbb {G}},widehat{varphi }) whenever the symbol x lies in L_{r,infty }(mathbb {G},varphi ), where 1/r=1/p-1/q. Moreover, we have ‖mx:Lp(G^,φ^)→Lq(G^,φ^)‖≤cp,q‖x‖Lr,∞(G,φ),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Vert m_{x}:L_p(\\widehat{\\mathbb {G}},\\widehat{\\varphi })\\rightarrow L_q(\\widehat{\\mathbb {G}},\\widehat{\\varphi })\\Vert \\le c_{p,q} \\Vert x\\Vert _{L_{r,\\infty }(\\mathbb {G},\\varphi )}, \\end{aligned}$$\\end{document}where c_{p,q} is a constant depending only on p and q. This was first proved by Hörmander (Acta Math 104:93–140, 1960) for mathbb {R}^n, and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of L_p-Fourier multipliers over discrete group von Neumann algebras. A similar result for mathcal {S}_p-mathcal {S}_q Schur multipliers is also proved.

Convergence of Fourier truncations for compact quantum groups and finitely generated groups

We generalize the Fejér-Riesz operator systems defined for the circle group by Connes and van Suijlekom to the setting of compact matrix quantum groups and their ergodic actions on C*-algebras. These truncations form filtrations of the containing C*-algebra. We show that when they and the containing C*-algebra are equipped with suitable quantum metrics, then under suitable conditions they converge to the containing C*-algebra for quantum Gromov-Hausdorff distance. Among other examples, our results are applicable to the quantum groups SUq(2) and their homogeneous spaces Sq2.