In the general theory of locally compact quantum groups, the notion of Haar measure (Haar weight) plays the most significant role. The aim of this paper is to carry out a careful analysis regarding Haar weight, in relation to general theory, for the specific non-compact quantum group (A,Δ) constructed earlier by the author. In this way, one can show that (A,Δ) is indeed a ``(C*-algebraic) locally compact quantum group'' in the sense of the recently developed definition given by Kustermans and Vaes. Attention will be given to pointing out the relationship between the original construction (obtained by deformation quantization) and the structure maps suggested by general theory.
The Haar measure on some locally compact quantum groups is constructed. The main example we treat is the az+b-group of Woronowicz. We also briefly consider some other examples (like the ax+b-group). We get the first examples of a locally compact quantum group where the Haar measure is not invariant with respect to the scaling group.
We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf \cst-algebras, and a similar map on the category of coinvolutive Hopf-von Neumann algebras. In the \cst-version, this functor sends $C_0(G)$ to $C^*(G)$ and vice versa, for every locally compact group $G$. As opposed to preceding approaches, there is an explicit description of commutative and co-commutative algebras in the range of this map (without assumption of being isomorph…
We develop the twisting construction for locally compact quantum groups. A new feature, in contrast to the previous work of M. Enock and the second author, is a non-trivial deformation of the Haar measure. Then we construct Rieffel's deformation of locally compact quantum groups and show that it is dual to the twisting. This allows to give new interesting concrete examples of locally compact quantum groups, in particular, deformations of the classical $az+b$ group and of the Woronowicz' quantum $az+b$ group.
The purpose of this paper is extend the notion of morphism of groupoids introduced by Zakrzewski to locally compact $σ$-compact groupoids endowed with Haar systems and to use the extension to construct a covariant functor from this category to a category of $C^{\ast}$-algebras in the sense of Woronowicz.
Given a locally compact group bundle, we show that the system of the Plancherel weights of their C*-algebras is lower semi-continuous. As a corollary, we obtain that the dual Haar sytem of a continuous Haar system of a locally compact abelian group bundle is also continuous.
The following paper is devoted to the study of type I locally compact quantum groups. We show how various operators related to the modular theory of the Haar integrals on $\mathbb{G}$ and $\widehat{\mathbb{G}}$ act on the level of direct integrals. Using these results we derive a web of implications between properties such as unimodularity or traciality of the Haar integrals. We also study in detail two examples: discrete quantum group $\widehat{\mathrm{SU}_q(2)}$ and the quantum $az+b$ group.
Idempotent states on a unimodular coamenable locally compact quantum group A are shown to be in one-to-one correspondence with right invariant expected C*-subalgebras of A. Haar idempotents, that is, idempotent states arising as Haar states on compact quantum subgroups of A, are characterised and shown to be invariant under the natural action of the modular element. This leads to the one-to-one correspondence between Haar idempotents on A and right invariant symmetric expected C*-subalgebras of A without the unimodularity assumption. Finally the tools developed in the first part of the paper a…
The paper is concerned with the extension of the classical study of probability measures on a compact group which are square roots of the Haar measure, due to Diaconis and Shahshahani, to the context of compact quantum groups. We provide a simple characterisation for compact quantum groups which admit no non-trivial square roots of the Haar state in terms of their corepresentation theory. In particular it is shown that such compact quantum groups are necessarily of Kac type and their subalgebras generated by the coefficients of a fixed two-dimensional irreducible corepresentation are isomorphi…
It is shown that a locally compact groupoid with open range map does not always admit a Haar system. It then is shown how to construct a Haar system if the stability groupoid and the quotient by the stability groupoid both admit one.
For a locally compact quantum group $\G$ with tracial Haar weight $φ$, and a quantum measure $μ$ on $\G$, we study the space ${H}_μ^p$ of $μ$-harmonic operators in the non-commutative $L^p$-space ${L}^p(\G)$ associated to the Haar weight $φ$. The main result states that if $μ$ is non-degenerate, then ${H}_μ^p$ is trivial for all $1 \leq p < \infty$.
We shall consider a locally compact groupoid endowed with a Haar system and having proper orbit space. We shall construct a groupoid C*-algebra which is independent of the Haar system (up to a *-isomorphism).
Correspondence between idempotent states and expected right-invariant subalgebras is extended to non-coamenable, non-unimodular locally compact quantum groups; in particular left convolution operators are shown to automatically preserve the right Haar weight.
We show that a compact quantum group all whose irreducible representations have dimension bounded by a fixed constant must be of Kac type, in other words, its Haar measure is a trace. The proof is based on establishing several facts concerning operators related to modular properties of the Haar measure. In particular we study spectrum of these operators and the dimension of some of their eigenspaces in relation to the quantum dimension of the corresponding irreducible representation.
In this paper we complete in several aspects the picture of locally compact quantum groups. First of all we give a definition of a locally compact quantum group in the von Neumann algebraic setting and show how to deduce from it a C*-algebraic quantum group. Further we prove several results about locally compact quantum groups which are important for applications, but were not yet settled in our paper "Locally compact quantum groups". We prove a serious strengthening of the left invariance of the Haar weight, and we give several formulas connecting the locally compact quantum group with its du…
The notion of Fourier transform is among the more important tools in analysis, which has been generalized in abstract harmonic analysis to the level of abelian locally compact groups. The aim of this paper is to further generalize the Fourier transform: Motivated by some recent works by Van Daele in the multiplier Hopf algebra framework, and by using the Haar weights, we define here the (generalized) Fourier transform and the inverse Fourier transform, at the level of locally compact quantum groups. We will then consider the analogues of the Fourier inversion theorem, Plancherel theorem, and t…
Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $α$ is a group homomorphism which pushes forward the Haar measure of $G$ to a measure absolutely continuous with respect to the Haar measure on $H$, then $(α\timesα)^{-1}$ preserves sets of compact operator synthesis, and conversely when $α$ is onto. We also prove similar preservation results for operator Ditkin sets and operator M-sets, obtaining preservation results for M-sets as corollaries. Some of these results extend or complement existing results of Ludwig, Shulman, Todorov a…
We give a necessary and sufficient condition on a compact semitopological quantum semigroup which turns it into a compact quantum group. In particular, we obtain a generalisation of Ellis's joint continuity theorem. We also investigate the question of the existence of the Haar state on a compact semitopological quantum semigroup and prove a "noncommutative" version of the converse Haar's theorem.
We find a combinatorial formula for the Haar functional of the orthogonal and unitary quantum groups. As an application, we consider diagonal coefficients of the fundamental representation, and we investigate their spectral measures.
Unlike for locally compact groups, idempotent states on locally compact quantum groups do not necessarily arise as Haar states of compact quantum subgroups. We give a simple characterisation of those idempotent states on compact quantum groups which do arise as Haar states on quantum subgroups. We also show that all idempotent states on the quantum groups U_q(2), SU_q(2), and SO_q(3) (q in (-1,0) \cup (0,1]) arise in this manner and list the idempotent states on the compact quantum semigroups U_0(2), SU_0(2), and SO_0(3). In the Appendix we provide a short new proof of coamenability of the def…
We construct a one parameter deformation of the group of $2\times 2$ upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way. Also, we give a complete description of the dual $\cs$-algebra and the dual comultiplication.
Generalizing the notion of matched pair of groups, we define and study matched pairs of locally compact groupoids endowed with Haar systems, in order to give new examples of measured quantum groupoids.
Idempotent states on locally compact quantum semigroups with weak cancellation properties are shown to be Haar states on a certain sub-object described by an operator system with comultiplication. We also give a characterization of the situation when this sub-object is actually a compact quantum subgroup. In particular we reproduce classical results on idempotent probability measures on locally compact semigroups with cancellation.
Let $G$ be second countable locally compact Hausdorff groupoid with a continuous Haar system. We remove the assumption of amenability in a theorem by Clark about GCR groupoid $C^*$-algebras. We show that if the groupoid $C^*$-algebra of $G$ is GCR then the orbits of $G$ are locally closed.
A notion of L^2-homology for compact quantum groups is introduced, generalizing the classical notion for countable, discrete groups. If the compact quantum group in question has tracial Haar state, it is possible to define its L^2-Betti numbers and Novikov-Shubin invariants/capacities. It is proved that these L^2-Betti numbers vanish for the Gelfand dual of a compact Lie group and that the zeroth Novikov-Shubin invariant equals the dimension of the underlying Lie group. Finally, we relate our approach to the approach of A. Connes and D. Shlyakhtenko by proving that the L^2-Betti numbers of a c…
Let $(G,α)$ and $(H,β)$ be locally compact Hausdorff groupoids with Haar systems, and let $(X,λ)$ be a topological correspondence from $(G,α)$ to $(H,β)$ which induce the ${C}^*$-correspondence $\mathcal{H}(X)\colon {C}^*(G,α)\to {C}^*(H,β)$. We give sufficient topological conditions which when satisfied the ${C}^*$-correspondence $\mathcal{H}(X)$ is proper, that is, the ${C}^*$-algebra ${C}^*(G,α)$ acts on the Hilbert ${C}^*(H,β)$-module ${H}(X)$ via the comapct operators. Thus a proper topological correspondence produces an element in ${KK}({C}^*(G,α),{C}^*(H,β))$.
We present a generalization of Hirschman's entropic uncertainty principle for locally compact abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups, and generalize the relation to compact quantum groups of Kac type. We also establish the complementarity of finite-dimensional quantum group algebras. In the non-unimodular setting, we obtain an uncertainty relation for arbitrary locally compact groups using the relative entropy with respect to the Haar weight as the measure of uncertainty. We also show that…
Let $\mathbb{G}$ be a locally compact quantum group with dual $\widehat{\mathbb{G}}$. Suppose that the left Haar weight $φ$ and the dual left Haar weight $\widehatφ$ are tracial, e.g. $\mathbb{G}$ is a unimodular Kac algebra. We prove that for $1<p\le 2 \le q<\infty$, the Fourier multiplier $m_{x}$ is bounded from $L_p(\widehat{\mathbb{G}},\widehatφ)$ to $L_q(\widehat{\mathbb{G}},\widehatφ)$ whenever the symbol $x$ lies in $L_{r,\infty}(\mathbb{G},φ)$, where $1/r=1/p-1/q$. Moreover, we have \begin{equation*} \|m_{x}:L_p(\widehat{\mathbb{G}},\widehatφ)\to L_q(\widehat{\mathbb{G}},\widehatφ)\|\l…
In this paper, we give an alternative approach to the theory of locally compact quantum groups, as developed by Kustermans and Vaes. We develop the theory completely within the von Neumann algebra framework. At various points, we also do things differently. We have a different treatment of the antipode. We obtain the uniqueness of the Haar weights in an early stage. We take advantage of this fact when deriving the other main results in the theory. We also give a slightly different approach to duality. Finally, we collect, in a systematic way, several important formulas. In an appendix, we indi…
Suppose G is a second countable, locally compact, Hausdorff, principal groupoid with a fixed left Haar system. We define a notion of integrability for groupoids, and show G is integrable if and only if the groupoid C*-algebra C*(G) has bounded trace.
If $G$ is a second countable locally compact Hausdorff groupoid with Haar system, we show that every representation induced from an irreducible representation of a stability group is irreducible.
We study the concept of co-amenability for a compact quantum group. Several conditions are derived that are shown to be equivalent to it. Some consequences of co-amenability that we obtain are faithfulness of the Haar integral and automatic norm-boundedness of positive linear functionals on the quantum group's Hopf *-algebra (neither of these properties necessarily holds without co-amenability).
Let G be second countable locally compact Hausdorff groupoid with a continuous Haar system. We remove the assumption of amenability in a theorem of Clark about groupoids whose $C^*$-algebras are CCR. We show that if the groupoid C*-algebra of G is CCR then the orbits of G are closed.
In the previous article, we proved that a topological correspondence $(X,λ)$ from a locally compact groupoid with a Haar system $(G,α)$ to another one, $(H,β)$, produces a $C^*$-correspondence $\mathcal{H}(X)$ from $C^*(G,α)$ to $C^*(H,β)$. In the present article, we describe how to form a composite of two topological correspondences when the bispaces are Hausdorff and second countable in addition to being locally compact.
This article extends the main results of the publication arXiv:2001.01312 to the case of a twisted groupoid. More precisely, it gives a decomposition of the C*-algebra of a twisted locally compact groupoid with Haar system in presence of a normal subgroupoid. When the normal subgroupoid and the twist over it are abelian, one obtains another twisted groupoid C*-algebra. This is applied to C*-algebraic deformation quantization and to multiplier representations of locally compact abelian groups.
We establish that a second countable locally compact groupoid possessing a continuous Haar system is topologically amenable if and only if it is Borel amenable. We give some examples and applications.
The theory of measured quantum groupoids, as defined by Lesieur and myself, was made to generalize the theory of quantum groups made by Kustarmans and Vaes, but was only defined in a von Neumann algebra setting; Th. Timmermann constructed locally compact quantum groupoids, which is a C*-version of quantum groupoids. Here, we associate to such a locally compact quantum groupoid a measured quantum groupoid in which it is weakly dense; we then associate to a measured quantum groupoid a locally compact quantum groupoid which is weakly dense in the measured quantum groupoid, but such a locally comp…
We introduce the construction of induced corepresentations in the setting of locally compact quantum groups and prove that the resulting induced corepresentations are unitary under some mild integrability condition. We also establish a quantum analogue of the classical bijective correspondence between quasi-invariant measures and certain measures on the larger locally compact group.
We prove that if $ρ: A(H) \to B(G)$ is a homomorphism between the Fourier algebra of a locally compact group $H$ and the Fourier-Stieltjes algebra of a locally compact group $G$ induced by a mixed piecewise affine map $α: G \to H$, then $ρ$ extends to a w*-w* continuous map between the corresponding $L^\infty$ algebras if and only if $α$ is an open map. Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $α$ is a group homomorphism which pushes forward the Haar measure of $G$ to a measure absolutely continuous with respect to the Haa…
We give some new characterizations of exactness for locally compact second countable groups. In particular, we prove that a locally compact second countable group is exact if and only if it admits a topologically amenable action on a compact Hausdorff space. This answers an open question by Anantharaman-Delaroche.
We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.
We review various notions of correspondences for locally compact groupoids with Haar systems, in particular a recent definition due to R.D. Holkar. We give the construction of the representations induced by such a correspondence. Finally, we extend the construction of induced representations to hypergroupoids.
Equivariant T-duality triples of locally compact abelian groups are considered. The motivating example dealing with the group $\R^n$ containing a lattice $\Z^n$ comes with an isomorphism in twisted equivariant K-theory.
In this short note we introduce a notion called "quantum injectivity" of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. Particularly, this provides a new characterization of amenability of locally compact groups.
In this paper, we generalize Young's inequality for locally compact quantum groups and obtain some results for extremal pairs of Young's inequality and extremal functions of Hausdorff-Young inequality.
It is proved that every second countable locally Hausdorff and locally compact continuous groupoid has a Borel set of units that meets every orbit and is what is called "lacunary," a property that implies that the intersection with every orbit is countable.
We extend the classical Stone duality between zero dimensional compact Hausdorff spaces and Boolean algebras. Specifically, we simultaneously remove the zero dimensionality restriction and extend to étale groupoids, obtaining a duality with an elementary class of inverse semigroups.
This is a short survey on idempotent states on locally compact groups and locally compact quantum groups. The central topic is the relationship between idempotent states, subgroups and invariant C*-subalgebras. We concentrate on recent results on locally compact quantum groups, but begin with the classical notion of idempotent probability measure. We also consider the `intermediate' case of idempotent states in the Fourier--Stieltjes algebra: this is the dual case of idempotent probability measures and so an instance of idempotent states on a locally compact quantum group.
We generalise the definition of a group algebra so that it makes sense for non-locally compact topological groups, in particular, we require that the representation theory of the group algebra is isomorphic (in the sense of Gelfand-Raikov) to the continuous representation theory of the group, or to some other important subset of representations. We prove that a group algebra if it exists, is always unique up to isomorphism. From examples, group algebras do not always exist for non-locally compact groups, but they do exist for some. We define a convolution on the dual of the Fourier-Stieltjes a…
