Induced Corepresentations of Locally Compact Quantum Groups

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.     

Duality and Self-Duality for Dynamical Quantum Groups

We define a natural concept of duality for the -Hopf algebroids introduced by Etingof and Varchenko. We prove that the special case of the trigonometric SL(2) dynamical quantum group is self-dual, and may therefore be viewed as a deformation both of the function algebra F(SL(2)) and of the enveloping algebra U(sl(2)). Matrix elements of the self-duality in the Peter–Weyl basis are 6j-symbols; this leads to a new algebraic interpretation of the hexagon identity or quantum dynamical Yang–Baxter equation for quantum and classical 6j-symbols.     

Geometrizing Infinite Dimensional Locally Compact Groups

We study groups having invariant metrics of curvature bounded below in the sense of Alexandrov. Such groups are a generalization of Lie groups with invariant Riemannian metrics, but form a much larger class. We prove that every locally compact, arcwise connected, first countable group has such a metric. These groups may not be (even infinite dimensional) manifolds. We show a number of relationships between the algebraic and geometric structures of groups equipped with such metrics. Many results do not require local compactness.

     

Continuity Criterion for Finite-Dimensional Representations of Locally Compact Groups

Mathematical Notes -     

Generalized Group Algebras of Locally Compact Groups

This article studies the homological properties of generalized group algebra L ¹(G, A) of a locally compact group G over a Banach algebra A with an identity of norm 1. It is shown that if L ¹(G, A) is right continuous then G is finite and A is right continuous. It is also shown that L ¹(G, A) is right self-injective if and only if G is finite and A is right self-injective.     

Gluing Compact Matrix Quantum Groups

We study glued tensor and free products of compact matrix quantum groups with cyclic groups – so-called tensor and free complexifications. We characterize them by studying their representation categories and algebraic relations. In addition, we generalize the concepts of global colourization and alternating colourings from easy quantum groups to arbitrary compact matrix quantum groups. Those concepts are closely related to tensor and free complexification procedures. Finally, we also study a more general procedure of gluing and ungluing.

     

Time Regularity for Random Walks on Locally Compact Groups

Let G be a compactly generated, locally compact group, and let T be the operator of convolution with a probability measure μ on G. Our main results give sufficient conditions on μ for the operator T to be analytic in L ^p (G), 1 < p < ∞, where analyticity means that one has an estimate of form for all n = 1, 2, ... in L ^p operator norm. Counterexamples show that analyticity may not hold if some of the conditions are not satisfied.     

Locally Compact Topological Groups and Cofinal Completeness

It is proved that a Tychonoff topological group is locally compactif and only if it is of pointwise countable type and its leftuniformity is cofinally complete. From this result a characterizationis derived of those T₀ paratopological groups (X, ) of pointwisecountable type for which (X, ^–1) is locally compactand also a characterization is deduced of locally pseudocompacttopological groups in terms of cofinal completeness. Also characterizedare the Tychonoff topological groups of pointwise countabletype for which their left uniformity has property U. Finally,cofinal completeness of the Hausdorff–Bourbaki uniformityof a topological group is studied.     

Totally Disconnected and Locally Compact Heisenberg-Weyl Groups

Harmonic analysis on ℤ(p ^ℓ ) and the corresponding representation of the Heisenberg-Weyl group HW[ℤ(p ^ℓ ),ℤ(p ^ℓ ),ℤ(p ^ℓ )], is studied. It is shown that the HW[ℤ(p ^ℓ ),ℤ(p ^ℓ ),ℤ(p ^ℓ )] with a homomorphism between them, form an inverse system which has as inverse limit the profinite representation of the Heisenberg-Weyl group \mathfrak HW[\mathbbZ_p,\mathbbZ_p,\mathbbZ_p]\mathfrak {HW}[{\mathbb{Z}}_{p},{\mathbb{Z}}_{p},{\mathbb{Z}}_{p}]. Harmonic analysis on ℤ _p is also studied. The corresponding representation of the Heisenberg-Weyl group HW[(ℚ _p /ℤ _p ),ℤ _p ,(ℚ _p /ℤ _p )] is a totally disconnected and locally compact topological group.     

The Unitary Implementation of a Locally Compact Quantum Group Action

In this paper we study actions of locally compact quantum groups on von Neumann algebras and prove that every action has a canonical unitary implementation, paralleling Haagerup's classical result on the unitary implementation of a locally compact group action. This result is an important tool in the study of quantum groups in action. We will use it in this paper to study subfactors and inclusions of von Neumann algebras. When α is an action of the locally compact quantum group (M, Δ) on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion N^αNM_αN is a basic construction. Here N^α denotes the fixed point algebra and M_αN is the crossed product. When α is an outer and integrable action on a factor N we prove that the inclusion N^αN is irreducible, of depth 2 and regular, giving a converse to the results of M. Enock and R. Nest (1996, J. Funct. Anal.137, 466–543; 1998, J. Funct. Anal.154, 67–109). Finally we prove the equivalence of minimal and outer actions and we generalize the main theorem of Yamanouchi (1999, Math. Scand.84, 297–319): every integrable outer action with infinite fixed point algebra is a dual action.     

A Class of Functional Equations on a Locally Compact Group

Let G be a locally compact group not necessarily unimodular.Let µ be a regular and bounded measure on G. We study,in this paper, the following integral equation, E(µ) This equation generalizes the functional equation for sphericalfunctions on a Gel'fand pair. We seek solutions in the spaceof continuous and bounded functions on G. If is a continuousunitary representation of G such that (µ) is of rank one,then tr((µ)(x)) is a solution of E(µ). (Here, trmeans trace). We give some conditions under which all solutionsare of that form. We show that E(µ) has (bounded and)integrable solutions if and only if G admits integrable, irreducibleand continuous unitary representations. We solve completelythe problem when G is compact. This paper contains also a listof results dealing with general aspects of E(µ) and propertiesof its solutions. We treat examples and give some applications.     

Ulyanov-Type Embedding Theorems for Functions on Zero-Dimensional Locally Compact Groups

Siberian Mathematical Journal - We establish some analogs of Ulyanov’s and Andrienko’s Theorems on the embedding of the Hölder spaces of integrable functions on zero-dimensional...     

The Derivation Problem for Group Algebras of Connected Locally Compact Groups

The derivation problem for a locally compact group G is to decidewhether for each derivation D from L¹(G) into L¹(G) there isa bounded measure µM(G) with D(a) = aµ–µa(a L¹(G)). In this paper we obtain an affirmative answer forthe case of connected groups. To explain the contents of thispaper we give an equivalent formulation of the problem. Supposethat the group G acts as a group of homeomorphisms of the locallycompact space X. Related to this there is an action of G onM(X). A bounded crossed homomorphism from G to M(X) is a map with bounded range and satisfying (gh) = g(h)+(g) (g, h G).The problem for bounded crossed homomorphisms is to decide iffor each such there is an element µ of M(X) with (g)= gµ– µ (g G). The derivation problem isequivalent to this bounded crossed homomorphism problem forthe special case X = G where G acts on X by conjugation (togetherwith some mild continuity hypotheses about the map :GM(X) whichare often automatically satisfied). The bounded crossed homomorphismproblem always has a positive solution if G is amenable anda closely related calculation shows that in solving the boundedcrossed homomorphism problem we need only solve it for functions which are zero on H where H is a given amenable subgroup ofG. It can happen that this condition of being zero on H forces to be zero even when H is a comparatively small subgroup ofG. If h is an element of G such that ‘hⁿx ’ asn for all x X then for any two measures µ and , forlarge values of n, µ and hⁿ have little overlap so ||µ+ hⁿ|| ||µ|| + ||||. Thus if H is the subgroup generatedby h, for any g G .     

Fundamental Sets of Functions on Locally Compact Abelian Groups

Let G be a locally compact Abelian group, and let X be a compact set of G. Given a positive definite function ?: G × G → ? whose real part is continuous at neutral element of G, we research a necessary and sufficient setting for the linear span of the set {x ∈ X → ?(x ? y): y ∈ X} to be dense in C(X) in the topology of uniform convergence. The context treated that is abstract encompasses classical cases of the literature, while other examples are entirely new.     

Duality and Multiplicative Stochastic Processes on Quantum Groups

An analogue of McKean's stochastic product integral is introduced and used to define stochastic processes with independent increments on quantum groups. The explicit form of the dual pairing (q-analogue of the exponential map) is calculated for a large class of quantum groups. The constructed processes are shown to satisfy generalized Feynman-Kac type formulas, and polynomial solutions of associated evolution equations are introduced in the form of Appell systems. Explicit calculations for Gauss and Poisson processes complete the presentation.