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Consider a locally compact quantum group G with a closed classical abelian subgroup Γ equipped with a 2-cocycle Ψ : Γ ^ × Γ ^ → C . We study in detail the associated Rieffel deformation G Ψ and establish a canonical correspondence between Γ -invariant convolution semigroups of states on G and on G Ψ . [ABSTRACT FROM AUTHOR]

We prove that an action ρ:A→M(C0(G)⊗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 G-equivariant compactifications: that the categories compactifications of ρ 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 G is regular coamenable we also show that the forgetful functor from unital G-C∗-algebras to unital C∗-algebras creates finite limits and is comonadic, and that the monomorphisms in the former category are injective. [ABSTRACT FROM AUTHOR]

Let and be Banach algebras, let be a Banach algebra epimorphism from to , and let be a nonzero character on . As is known if is -contractible (amenable) then is -contractible (amenable). We prove that the converse is true under some conditions. As an important application, we study the -contractibility and amenability of the convolution algebra of trace class operators , where is a locally compact quantum group, and is a nonzero character on . [ABSTRACT FROM AUTHOR]

Let 풜 be a semiprime Banach algebra and let φ be a nonzero character on 풜. In this paper, we discuss the projectivity of certain left Banach 풜-modules in terms of left φ-contractibility of Banach algebras. We also apply our results for the category of quantum group algebras and eventually, we present illustrative examples of the projective left Banach modules over the group algebra L1(G) and the Fourier algebra A(G) of a locally compact group G. [ABSTRACT FROM AUTHOR]

Let be a locally compact quantum group. Then the space T (L 2 ()) of trace class operators on L 2 () is a Banach algebra with the convolution induced by the right fundamental unitary of . We show that properties of such as amenability, triviality and compactness are equivalent to the existence of left or right invariant means on the convolution Banach algebra T (L 2 ()). We also investigate the relation between the existence of certain (weakly) compact right and left multipliers of T (L 2 ()) ∗ ∗ and some properties of . [ABSTRACT FROM AUTHOR]

We prove that discrete compact quantum groups (or more generally locally compact, under additional hypotheses) with coamenable dual are continuous fields over their central closed quantum subgroups, and the same holds for free products of discrete quantum groups with coamenable dual amalgamated over a common central subgroup. Along the way we also show that free products of continuous fields of C ∗ -algebras are again free via a Fell-topology characterization for C ∗ -field continuity, recovering a result of Blanchard's in a somewhat more general setting. [ABSTRACT FROM AUTHOR]

We make a careful study of one-parameter isometry groups on Banach spaces, and their associated analytic generators, as first studied by Cioranescu and Zsido. We pay particular attention to various, subtly different, constructions which have appeared in the literature, and check that all give the same notion of generator. We give an exposition of the “smearing” technique, checking that ideas of Masuda, Nakagami and Woronowicz hold also in the weak∗-setting. We are primarily interested in the case of one-parameter automorphism groups of operator algebras, and we present many applications of the machinery, making the argument that taking a structured, abstract approach can pay dividends. A motivating example is the scaling group of a locally compact quantum group G and the fact that the inclusion C0(G) → L∞(G) intertwines the relevant scaling groups. Under this general setup, of an inclusion of a C*-algebra into a von Neumann algebra intertwining automorphism groups, we show that the graphs of the analytic generators, despite being only non-self-adjoint operator algebras, satisfy a Kaplansky Density style result. The dual picture is the inclusion L¹ (G) → M(G), and we prove an “automatic normality” result under this general setup. The Kaplansky Density result proves more elusive, as does a general study of quotient spaces, but we make progress under additional hypotheses. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the theory of induced representations in the setting of locally compact quantum groups. In the case of induction from open quantum subgroups, we show that constructions of Kustermans and Vaes are equivalent to the classical, and much simpler, construction of Rieffel. We also prove in general setting the continuity of induction in the sense of Vaes with respect to weak containment. [ABSTRACT FROM AUTHOR]

We study lattice operations on the set of idempotent states on a locally compact quantum group corresponding to the operations of intersection of compact subgroups and forming the subgroup generated by two compact subgroups. Normal (σ-weakly continuous) idempotent states are investigated and a duality between normal idempotent states on a locally compact quantum group G and on its dual pG is established. Additionally we analyze the question of when a left coideal corresponding canonically to an idempotent state is finite-dimensional and give a characterization of normal idempotent states on compact quantum groups. [ABSTRACT FROM AUTHOR]

For locally compact quantum groups 𝔾 , we initiate an investigation of stable states with respect to unitary co-representations U of 𝔾 on Hilbert spaces H U ; in particular, we study the subject on the multiplicative unitary operator W 𝔾 of 𝔾 with some examples on locally compact quantum groups arising from discrete groups and compact groups. As the main result, we consider the one co-dimensional Hilbert subspace of H U associated to a suitable vector η , to present an operator theoretic characterization of stable states with respect to a related unitary co-representation U η . This provides a quantum version of an interesting result on unitary representations of locally compact groups given by Lau and Paterson in 1991. [ABSTRACT FROM AUTHOR]

Abstract 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 Sołtan, 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. [ABSTRACT FROM AUTHOR]

Abstract The subject of this paper is the study of convolution semigroups of states on a locally compact quantum group, generalising classical families of distributions of a Lévy process on a locally compact group. In particular a definitive one-to-one correspondence between symmetric convolution semigroups of states and noncommutative Dirichlet forms satisfying the natural translation invariance property is established, extending earlier partial results and providing a powerful tool to analyse such semigroups. This is then applied to provide new characterisations of the Haagerup Property and Property (T) for locally compact quantum groups, and some examples are presented. The proofs of the main theorems require developing certain general results concerning Haagerup's L p -spaces. [ABSTRACT FROM AUTHOR]

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 ⁎-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. [ABSTRACT FROM AUTHOR]

We initiate a study of operator approximate biprojectivity for quantum group algebra L¹(G), where G is a locally compact quantum group. We show that if L¹(G) is operator approximately biprojective, then G is compact. We prove that if G is a compact quantum group and H is a non-Kac-type compact quantum group such that both L¹(G) and L¹(H) are operator approximately biprojective, then L¹(G) ⊗ L¹(H) is operator approximately biprojective, but not operator biprojective. [ABSTRACT FROM AUTHOR]

In this short note, we further Ng's work by extending Bekka amenability and weak Bekka amenability to general locally compact quantum groups, and we generalize some of Ng's results to the general case. In particular, we show that a locally compact quantum group G is coamenable if and only if the contra-corepresentation of its fundamental multiplicative unitary WG is Bekka-amenable, and that G is amenable if and only if its dual quantum group's fundamental multiplicative unitary WĜ is weakly Bekka-amenable. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For a locally compact quantum group [ABSTRACT FROM AUTHOR]

The notion of an open quantum subgroup of a locally compact quantum group is introduced and given several equivalent characterizations in terms of group-like projections, inclusions of quantum group C ⁎ -algebras and properties of respective quantum homogeneous spaces. Open quantum subgroups are shown to be closed in the sense of Vaes and normal open quantum subgroups are proved to be in 1–1 correspondence with normal compact quantum subgroups of the dual quantum group. [ABSTRACT FROM AUTHOR]

Given a Banach algebra $${{\mathcal{A}}}$$ , for a non-zero character $${\varphi}$$ on $${{\mathcal{A}}}$$ , we characterize the existence of $${\varphi}$$ -means on a left introverted subspace of $${{\mathcal{A}^{*}}}$$ containing $${\varphi}$$ in terms of certain derivations from $${{\mathcal{A}}}$$ into certain Banach $${{\mathcal{A}}}$$ -bimodules. We also adapt and extend a result in (Crann and Neufang, Trans Amer Math Soc 368:495-513, 2016) on locally compact quantum groups to the Banach algebra setting which, in particular, answers a question of Bédos and Tuset, concerning the amenability of locally compact quantum groups. [ABSTRACT FROM AUTHOR]

We introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study their properties. Our approach is slightly different from the one introduced in the thesis of S. Roy [16] and, unlike the investigations of Roy, we work within the von Neumann algebraic picture. This allows the use of powerful techniques related to crossed products by actions of locally compact quantum groups. In particular we show existence of a “braided comultiplication” on the algebra spanned by slices of “braided multiplicative unitary”. [ABSTRACT FROM AUTHOR]

In this short paper, we obtained some equivalent formulations of property T for a general locally compact quantum group 픾, in terms of the full quantum group C*-algebras and the *-representation of associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if 픾 is of Kac type, we show that 픾 has property T if and only if every finite-dimensional irreducible *-representation of is an isolated point in the spectrum of (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. [ABSTRACT FROM AUTHOR]

We consider induced coactions on C*-algebras along a homomorphism of locally compact quantum groups which need not give a closed quantum subgroup. Our approach generalizes the induced coactions constructed by Vaes, and also includes certain fixed point algebras. We focus on the case when the homomorphism satisfies a quantum analogue of properness. Induced coactions along such a homomorphism still admit the natural formulations of various properties known in the case of a closed quantum subgroup, such as imprimitivity and adjointness with restriction. Also, we show a relationship of induced coactions and restriction which is analogous to base change formula for modules over algebras. As an application, we give an example that shows several kinds of 1-categories of coactions with forgetful functors cannot recover the original quantum group. [ABSTRACT FROM AUTHOR]

For a locally compact quantum group 픾, we generalize some notions of amenability such as amenability of locally compact quantum groups and inner amenability of locally compact groups to the case of right Banach L1(픾)-modules. Also, we investigate the concept of harmonic functionals over right Banach L1(픾)-modules and use these devices to study, among others, amenability of 픾. [ABSTRACT FROM AUTHOR]

Abstract: We discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups. On the von Neumann algebra level we recall an interesting duality for such objects studied earlier by M. Izumi, R. Longo, S. Popa for compact Kac algebras and by M. Enock in the general case of locally compact quantum groups. A definition of a quantum homogeneous space is proposed along the lines of the pioneering work of Vaes on induction and imprimitivity for locally compact quantum groups. The concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces. Among various examples we single out the quantum analog of the quotient of the Cartesian product of a quantum group with itself by the diagonal subgroup, analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles. The former turns out to be an interesting application of the duality mentioned above. [Copyright &y& Elsevier]

We initiate a study of inner amenability for a locally compact quantum group 픾 in the sense of Kustermans-Vaes. We show that all amenable and co-amenable locally compact quantum groups are inner amenable. We then show that inner amenability of 픾 is equivalent to the existence of certain functionals associated to characters on L1(픾). For co-amenable locally compact quantum groups, we introduce and study strict inner amenability and its relation to the extension of the co-unit ϵ from C0(픾) to L∞(픾). We then obtain a number of equivalent statements describing strict inner amenability of 픾 and the existence of certain means on subspaces of L∞(픾) such as LUC(픾), RUC(픾) and UC(픾). Finally, we offer a characterization of strict inner amenability in terms of a fixed point property for L1(픾)-modules. [ABSTRACT FROM AUTHOR]

A natural scheme is established for the approximation of quantum Lévy processes on locally compact quantum groups by quantum random walks. We work in the somewhat broader context of discrete approximations of completely positive quantum stochastic convolution cocycles on C*-bialgebras. [ABSTRACT FROM AUTHOR]

We show that any completely positive multiplier of the convolution algebra of the dual of an operator algebraic quantum group G (either a locally compact quantum group, or a quantum group coming from a modular or manageable multiplicative unitary) is induced in a canonical fashion by a unitary corepresentation of G. It follows that there is an order bijection between the completely positive multipliers of L¹ (G) and the positive functionals on the universal quantum group C0″(G). We provide a direct link between the Junge, Neufang, Ruan representation result and the representing element of a multiplier, and use this to show that their representation map is always weak*-weak*-continuous. [ABSTRACT FROM AUTHOR]

Abstract: Let and be Woronowiczʼs q-deformations of respectively the compact Lie group and the non-trivial double cover of the Lie group of Euclidean transformations of the plane. We prove that, in some sense, their duals are ‘Morita equivalent locally compact quantum groups’. In more concrete terms, we prove that the von Neumann algebraic quantum groups and are unitary cocycle deformations of each other. [Copyright &y& Elsevier]

Abstract: Beginning with a skew-symmetric matrix, we define a certain Poisson–Lie group. Its Poisson bracket can be viewed as a cocycle perturbation of the linear (or “Lie–Poisson”) Poisson bracket. By analyzing this Poisson structure, we gather enough information to construct a -algebraic locally compact quantum group, via the “cocycle bicrossed product construction” method. The quantum group thus obtained is shown to be a deformation quantization of the Poisson–Lie group, in the sense of Rieffel. [Copyright &y& Elsevier]

Abstract: We show that there is a one-to-one correspondence between compact quantum subgroups of a co-amenable locally compact quantum group and certain left invariant -subalgebras of . We also prove that every compact quantum subgroup of a co-amenable quantum group is co-amenable. Moreover, there is a one-to-one correspondence between open subgroups of an amenable locally compact group G and non-zero, invariant -subalgebras of the group -algebra . [Copyright &y& Elsevier]

Let G be a locally compact abelian group with dual group Ĝ. The Hausdorff-Young theorem states that if f ∈ Lp(G), where 1 ≤ p ≤ 2, then its Fourier transform ${\mathcal F}_p(f)$ belongs to Lq(Ĝ) (where (1/p) + (1/q) = 1) and $\Vert{\mathcal F}_p(f)\Vert_q \leq \Vert{f}\Vert_p$. Kunze and Terp extended this to unimodular and locally compact groups, respectively. We further generalize this result to an arbitrary locally compact quantum group 픾 by defining a Fourier transform ${\mathcal F}_p:L_p(\mathbb G) \to L_q(\hat{\mathbb G})$ and showing that this Fourier transform satisfies the Hausdorff-Young inequality. [ABSTRACT FROM AUTHOR]

Abstract: For a Banach algebra A with a bounded approximate identity, we investigate the A-module homomorphisms of certain introverted subspaces of , and show that all A-module homomorphisms of are normal if and only if A is an ideal of . We obtain some characterizations of compactness and discreteness for a locally compact quantum group . Furthermore, in the co-amenable case we prove that the multiplier algebra of can be identified with . As a consequence, we prove that is compact if and only if and ; which partially answer a problem raised by Volker Runde. [Copyright &y& Elsevier]

We define and study square-integrable coactions of locally compact quantum groups on Hilbert modules, generalising previous work for group actions. As special cases, we consider square-integrable Hilbert space corepresentations and integrable coactions on C *-algebras. Our main result is an equivariant generalisation of Kasparov''s Stabilisation Theorem. [Copyright &y& Elsevier]

Recently, Neufang, Ruan and Spronk proved a completely isometric representation theorem for the measure algebra M(G) and for the completely bounded (Herz–Schur) multiplier algebra McbA(G) on $\mathcal{B}(L_{2}(G))$, where G is a locally compact group. We unify and generalize both results by extending the representation to arbitrary locally compact quantum groups 픾 = (M, Γ, φ, ψ). More precisely, we introduce the algebra $M_{\rm cb}^{r} (L_1(\mathbb{G}))$ of completely bounded right multipliers on L1(픾) and we show that $M^r_{\rm cb} (L_1(\mathbb{G}))$ can be identified with the algebra of normal completely bounded $\hat{M}$-bimodule maps on $\mathcal{B}(L_2(\mathbb{G}))$ which leave the subalgebra M invariant. From this representation theorem, we deduce that every completely bounded right centralizer of L1(픾) is in fact implemented by an element of $M_{\rm cb}^r (L_1(\mathbb{G}))$. We also show that our representation framework allows us to express quantum group "Pontryagin" duality purely as a commutation relation. [ABSTRACT FROM AUTHOR]