Harmonic Analysis of Probability Measures on Hypergroups 9783110877595, 9783110121056

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de Gruyter Studies in Mathematics 20 Editors: Heinz Bauer · Jerry L. Kazdan · Eduard Zehnder

de Gruyter Studies in Mathematics 1 Riemannian Geometry, Wilhelm Klingenberg 2 Semimartingales, Michel Metivier 3 Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup 4 Spaces of Measures, Corneliu Constantinescu 5 Knots, Gerhard Burde and Heiner Zieschang 6 Ergodic Theorems, Ulrich Krengel 7 Mathematical Theory of Statistics, Helmut Strasser 8 Transformation Groups, Tammo torn Dieck 9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii 10 Analyticity in Infinite Dimensional Spaces, Michel Herve 11 Elementary Geometry in Hyperbolic Space, Werner Fenchel 12 Transcendental Numbers, Andrei B. Shidlovskii 13 Ordinary Differential Equations, Herbert Amann 14 Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and Francis Hirsch 15 Nevanlinna Theory and Complex Differential Equations, Ilpo Laine 16 Rational Iteration, Norbert Steinmetz 17 Korovkin-type Approximation Theory and its Applications, Francesco Altomare and Michele Campiti 18 Quantum Invariants of Knots and 3-Manifolds, Vladimir G. Turaev 19 Dirichlet Forms and Symmetric Markov Processes, Masatoshi Fukushima, Yoichi Oshima, Mas ay o shi Take da

Walter R. Bloom · Herbert Heyer

Harmonic Analysis of Probability Measures on Hypergroups

w

Walter de Gruyter G Berlin · New York 1995 DE

Authors Walter R. Bloom School of Mathematical and Physical Sciences Murdoch University Perth, Western Australia 6150 Australia

Series Editors Heinz Bauer Mathematisches Institut der Universität Bismarckstraße ' D-91054 Erlangen, FRG

Herbert Heyer Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 D-72076 Tübingen Germany

Jerry L. Kazdan Department of Mathematics University of Pennsylvania 209 South 33rd Street Philadelphia, PA 19104-6395, USA

Eduard Zehnder ETH-Zentrum/Mathematik Rämistraße 101 CH-8092 Zürich Switzerland

7997 Mathematics Subject Classification: 43-02; 43A62, 43A10, 60B15, 60B99, 31C99, 60B10, 28A33, 43A15, 43A30, 43A35 Keywords: Hypergroups, probability measures, harmonic analysis

) Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Bloom, Walter R., 1948Harmonic analysis of probability measures on hypergroups / Walter R. Bloom, Herbert Heyer. p. cm. - (De Gruyter studies in mathematics ; 20) Includes bibliographical references and index. ISBN 3-11-012105-0 1. Hypergroups. 2. Harmonic analysis. 3. Probability measures. I. Heyer, Herbert. II. Title. III. Series. QA174.2.B58 1995 512'.55-dc20 94-37375

Die Deutsche Bibliothek — CIP-Einheitsaufnahme Bloom, Walter R.:

Harmonic analysis of probability measures on hypergroups / Walter R. Bloom ; Herbert Heyer. - Berlin ; New York : de Gruyter, 1994 (De Gruyter studies in mathematics ; 20) ISBN 3-11-012105-0 NE: Heyer, Herbert:; GT

© Copyright 1994 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typesetting: ASCO Trade Ltd. Hong Kong. Printing: Gerike GmbH, Berlin. Binding: Dieter Micolai, Berlin. Cover design: Rudolf Hübler, Berlin.

Contents

Introduction

1

1 Hypergroups and their measure algebras

6

1.1 Definition and general constructions 1.2 Translation and convolution 1.3 Invariant measures 1.4 Convolution of functions 1.5 Subhypergroups and double coset hypergroups 1.6 Idempotent measures and multipliers Notes

6 15 28 41 49 61 69

2 The dual of a commutative hypergroup

74

2.1 Representations and Fourier transforms 2.2 The dual space in the commutative case 2.3 Modification of the convolution 2.4 The dual hypergroup 2.5 Support of the Plancherel measure Notes

74 78 101 110 120 128

3 Some special classes of hypergroups

133

3.1 Polynomial hypergroups in several variables 3.2 Polynomial hypergroups in one variable 3.3 Examples of polynomial hypergroups in one variable 3.4 One-dimensional hypergroups 3.5 Sturm-Liouville hypergroups 3.6 Characterization of Pontryagin hypergroups Notes

133 147 163 189 201 248 255

4 Positive and negative definite functions and measures

258

4.1 Positive definite functions 4.2 The Levy continuity theorem 4.3 Positive definite measures 4.4 Negative definite functions 4.5 The Levy-Khintchine representation Notes

258 276 283 292 304 313

vi

Contents

5 Convolution semigroups and divisibility of measures

317

5.1 Convergence of nets of measures 5.2 Convolution semigroups of measures 5.3 Embedding infinitely divisible measures 5.4 Factorization on hermitian hypergroups Notes

317 329 348 355 379

6 Transience of convolution semigroups

382

6.1 The dichotomy theorem for random walks 6.2 The generalized Chung-Fuchs criterion 6.3 Transience and renewal of convolution semigroups 6.4 Characterization of potential measures 6.5 Invariant Dirichlet forms Notes

382 393 400 409 421 443

7 Randomized sums of hypergroup-valued random variables

446

7.1 Concretization of hypergroups 7.2 Moment functions 7.3 Strong laws of large numbers 7.4 Central limit theorems 7.5 Invariance principles Notes

446 452 468 487 515 530

8 Further topics

534

8.1 Towards a structure theory for hypergroups 8.2 Towards a theory of stationary random fields over hypergroups

534 546

Bibliography Examples Symbols Index

553 581 589 597

Introduction

Both harmonic analysis and probability theory on hypergroups are fields of research in rapid development. There is a fundamental impetus for extending harmonic analysis and representation theory beyond the class of locally compact semigroups, and there exists an urgent demand for studying stochastic dynamical systems within the framework of algebraic-topological structures specified by invariance conditions. One of the decisive steps in pursuing these two branches of investigation was the rediscovery of the notion of hypergroup which in fact dates back to the time of the rise of group theory with the work of Frobenius around 1900. As autonomous algebraic objects hypergroups were studied by F. Marty and M.S. Wall in the mid thirties, mainly within the theory of nonabelian groups and the related structures of spaces of conjugacy classes and double cosets. Understandably at that time no generally recognized definition was available. It took efforts emerging from various fields of mathematics, from algebra to mathematical physics, to formulate an axiomatic setting that together with some weaker versions such as hypercomplex systems and signed hypergroups meets the requirements of a suitable theoretical foundation. In fact the axiomatic access to the theory has been inspired and kept justified by progress in the study of Hecke algebras (Krieg (1990)) and quantum Gelfand pairs (Koornwinder (1991)). In our monograph, which is designed to present in a systematic way the applications of the hypergroup method to problems in probability theory, we shall deal exclusively with topological hypergroups, leaving aside the precious finite hypergroups, and focus on those that are commutative. To be a little more precise we shall consider hypergroups as locally compact spaces with a group-like structure on which the bounded measures convolve in a similar way to that on a locally compact group. Important examples of such hypergroups are double coset spaces, spaces of conjugacy classes, orbit spaces and duals arising from certain classes of locally compact groups, and corresponding topological group actions; but also the spaces Z+, IR+ of nonnegative integers and reals respectively, the unit interval D and the unit disc, with indeterministic operations different from the usual (deterministic) ones inherited from the group operations in Z, R and the complex numbers C respectively. In fact a hypergroup K can be viewed as a probabilistic group in the sense that to each pair x, y of points in K there exists a probability measure εχ * ey on Κ with compact support, not necessarily equal to the Dirac measure ε,.,, for a composition χ · y in K, such that (x,y) -»supp^ * e y ) is a continuous mapping from Κ χ Κ into the space of compact subsets of K. The convolution * between Dirac measures extends to all bounded measures on K and shifts the algebraic-topological analysis from the sparsely structured base space K to the generalized measure algebra Mb(K) of K. In place of the natural left translation of a function / by x, available in the group case, we deal in a hypergroup with a

2

Introduction

generalized (left) translation defined by T'f(y):=

f

JK

f(z)(ex*ey)(dz)

for all y e K. The generalized translation operator Tx had been introduced by Delsarte (1938) in connection with an extension of Taylor's formula, and applied by Levitan (1945) to second order linear differential equations. During the following decade Bochner (1954, 1956) employed the idea of generalized translations in his studies of the heat equation associated with Bessel and Gegenbauer eigenfunctions. Subsequent work of Dunkl (1973), Jewett (1975) and Spector (1975) has given rise to a mature axiomatic framework for generalized translation spaces, convolution algebras and hypergroups, and this has gained remarkable appreciation across a wide spectrum of mathematics. For commutative hypergroups K a substantial body of harmonic analysis based on Gelfand's theory of normed algebras has been built up. Since K admits a translation-invariant (Haar) measure ωκ the hypergroup algebra Ll(K) := Ll(K, ωκ) and related function algebras become the key objects of study. Despite the deficiency that the dual space of a hypergroup does not necessarily carry a hypergroup structure, Fourier and Plancherel transforms are available as important technical tools, and some duality theory can be established including the theory of positive and negative definite functions. What can hardly be expected at this stage is a structure theory for hypergroups in the spirit of the Pontryagin-van K mpen theorem. On the other hand the construction and analysis of all hypergroup structures on IR+ seems to be a realistic goal as a first step in this direction. Let us look at a sample hypergroup structure on R+ that arises when considering the Gelfand pair (G,H), where G is the motion group M(d) of Rd, and H the subgroup of rotations about a fixed point. Then clearly G/H coincides with Rd, the orbits in G/H with respect to the action of H form a family of concentric spheres, and each element of the double coset hypergroup K = G//H is determined by the radius of the corresponding sphere. As a consequence K can be identified with IR+, and R+ inherits its convolution structure from the group M(d). It turns out that the characters of K are the modified Bessel functions, and hence the hypergroup structure on R+ is obviously related to a Bessel eigenvalue problem. However this problem together with even more general Sturm-Liouville eigenvalue problems can be solved also for non-integer "dimensions" which indicates in more precise terms Levitan's original motivation for his contributions to the theory. Looking at more general pairs (G, H) the hypergroup view provides an extra insight into centres of group and measure algebras of locally compact groups G. For any relatively compact subgroup B of Aut(G) with B => Int(G) the set ZBMb(G) := (μ e Mb(G): μ ο β = μ for all β e Β}

is a commutative Banach subalgebra of Μb(G). Clearly ZBMb(G) coincides with the centre of Mb(G) whenever Β = Int(G). In the general situation we consider the space

Introduction

3

GB of -orbits of G. It is easily seen that (as Banach spaces) ZBMb(G) and Mb(GB) coincide. Moreover we have the identification ZBMb(G) =; Mb(G'//H) with G' the semidirect product G © B, and H = {e} χ Β where the canonical mapping G'//H -> GB is given by H(x, β)Η -» (x) for all (x, )eG xB. These two identifications show that ZBMb(G) can be studied without explicit reference to G but rather solely in terms of the derived hypergroups GB and G'//H. With the hypergroup convolutions in GB and G'//H the above identifications are in fact Banach algebra isomorphisms. In the special case that G := Ud and B := S0(d) the algebra ZBMb(G) consists of the radial measures on Ud. It becomes apparent that the space GB is homeomorphic to R+, and that R + carries the Bessel-Kingman structure originating from the motion group M(d). In the present monograph we attempt to give a balanced account of the theory of convolution semigroups on a (commutative) hypergroup K, or equivalently of stationary Markov processes with independent increments in K, along with the necessary harmonic analysis needed to describe these objects. We first provide the necessary background information in Chapters 1 to 3 and then devote Chapters 4 to 7 to the elaboration of the subject proper. More precisely in Chapters 1 to 3 we give the elementary analysis of hypergroups, their duality properties and the main constructions of examples with the emphasis in Chapter 3 on discrete hypergroups (with base space Z + ) arising from a linearization of orthogonal polynomials, and on one-dimensional hypergroups (with base space IR+ or I) arising from product (addition) formulae valid for the solutions of Sturm-Liouville eigenvalue problems. With such an extensive list of examples we have found it convenient to give a summary list which appears immediately following the bibliography. The main aspects of structural probability theory treated in Chapters 4 to 7 are divided into those (Chapters 4 and 5) that can be presented mainly in functional-analytic terms (representation of convolution semigroups and infinitely divisible measures via negative definite functions) and those (Chapters 6 and 7) where the discussion requires a genuinely probabilistic approach (asymptotic theory of families of random variables taking values in the given hypergroup). To maintain the balance of the exposition we add two appendices on recent research, towards a structure theory for hypergroups and towards a theory of stationary random fields over hypergroups respectively. A glance into the study of isotropic random walks shows the impact of the hypergroup method on the theory of Markov chains. We consider a two-point homogeneous metric space (£, d) with isometry group G. If H denotes the stabilizer of some point x0 of £ then G/H is identifiable with £, and G//H with the space of //-orbits of £ which by the two-point homogeneity is identifiable with the subset D := (d(x,x0): χ 6 £} of R+. By this very construction D carries a unique commutative hypergroup structure. Isotropic random walks on £ are stationary Markov chains on £ starting at x0 with G-invariant transition probabilities. They can be studied most efficiently in terms of //-invariant random walks on G with starting distribution ωΗ and transition probabilities (μ*εχ)(Β) for some Η-invariant measure μ on G, or in terms of random walks on F = G//H with starting distribution ε0 and transition probabilities ρ(μ) * ε, where ρ denotes the canonical projection G -» F.

4

Introduction

More generally the hypergroup method yields a direct approach to (additive) processes (Sn: n e Z+) with independent increments on a hypergroup K with transition probabilities given by the convolutions μηη*εχ(Β) for some two-parameter family {//„,,„: m < n} of probability measures on K. The interpretation of the increment between the K-valued random variables Sm and S„ requires a definition of the randomized sum Sm+l + · · · + Sn which in absence of any deterministic operation in K will depend on auxiliary probability measures for each pair of summands. This randomized addition of K-valued random variables has been introduced by Haldane (1960) and Kingman (1963) in such a way that as in the classical case the distribution of the sum of two independent K-valued random variables equals the convolution in Mb(K) of the distributions of the summands. We easily recognize random walks ( S n : n e / + ) with starting distribution μ on Κ as additive processes (with stationary independent increments) on Κ for which the measures //m „ are just the (n — m)-fold convolution powers of μ. Since randomized addition (of K-valued random variables) is neither associative nor distributive the classical probabilistic analysis cannot be applied. It necessitates the notion of moment functions to be constructed universally for large classes of hypergroups Κ to transform additive processes on Κ into real-valued martingales, thereby overcoming the above mentioned obstacles and leading to considerable progress in the theory. In visualizing the extent of the theory of hypergroups, its interrelations with other fields of mathematics and its wealth of applications outside mathematics proper we might venture a comparison with the classical areas of Lie groups or Banach spaces. Evidently the theory of hypergroups is less developed than those traditional structures. It is certainly subject to individual judgement in which direction the theory and applications of hypergroups should be advanced. In this book our choice has been to emphasize the harmonic analysis of probability measures, and our intention has been to guide workers in the field on a smoothly scaled ladder from the foundation stones towards the current state of the edifice. The monograph in hand grew out of intensive collaboration of the authors over a period of more than ten years. The second named author lectured on the subject for the first time at Hokkaido University in 1985 where he spent three months as a Fellow of the Japan Society for the Promotion of Sciences, and later to graduate students at his home university. The authors started their work on the manuscript in 1987 when the first named spent a research year as a Fellow of the Humboldt Foundation at the University of T bingen. Both authors acknowledge the generous support that the Foundation provided to its Fellow and his host, both in 1987 and on a second occasion in 1991. The first named author also wishes to acknowledge the extensive financial support granted to him by Murdoch University through study leave, and conference and special research grants. His coauthor owes gratitude to the Kultusministerium der Landersregierung von Baden-W rttemberg for sabbatical leaves in 1989 and 1993 which he spent as Visiting Professor at the University of California at San Diego where he found the solitude necessary for working on the book. It goes without mention but it ought to be mentioned that the book could hardly have been completed without the valuable contributions and the efficient help of several friends and colleagues. We would like to name in particular J rgen Gaiser,

Introduction

5

Stefan Kastner, Christian Rentzsch, Michael Voit, Richard Vrem and Hansmartin Zeuner who read parts of the manuscript and made helpful comments, and Nicola Armstrong and Xu Zengfu who contributed to the preparation of the bibliography. Last but by no means least we are grateful to Pamela Anthony at the School of Mathematical and Physical Sciences of Murdoch University for her expertise, skill and endurance in producing the typescript. Finally we give sincere thanks to Professor Dr. Heinz Bauer for his invitation to write this book for the prestigious series "Studies in Mathematics", and to Dr. Manfred Karbe of Walter de Gruyter Publishers for his patience and cooperation in its production. Perth and Tübingen July 1994

Walter R. Bloom and Herbert Heyer

Chapter 1

Hypergroups and their measure algebras

We begin in Section 1 by defining a hypergroup, which is a locally compact Hausdorff space K together with a convolution on its (bounded) measure space Mb(K] such that (Mb(K\ +, *) is a Banach algebra with unit, and having further convolution-like properties. Standard examples including the space of conjugacy classes of a compact group, the space of double cosets of a locally compact group, and the dual object of a compact group will be presented in Section 2. (Later, in Chapter 3, we will study in detail two major classes of hypergroups, namely polynomial hypergroup structures on the nonnegative integers, and hypergroups on the half line.) In Section 3 we introduce subinvariant and invariant measures, and outline Spector's deep result on the existence of invariant (Haar) measure for commutative hypergroups. We are then able to introduce the convolution of two functions, and the study of properties of this is the subject of Section 4. Section 5 is concerned with subhypergroups and double coset hypergroups, paralleling the corresponding development for locally compact groups, and ends with an introduction to hypergroup joins. Finally in Section 6 we consider a Cohen-type result on the classification of idempotent measures, and also present some results on bounded approximate units and multipliers for Ll(K).

1.1 Definition and general constructions Before we present the definition of a hypergroup we need to set the scene with properties of Radon measures on a locally compact space, and especially the various associated topologies. Many of these details are well known, but it is convenient to collect them together here for ease of future reference. 1.1.1 Notation and preliminaries. Throughout this section X will denote a locally compact Hausdorff space. Functions. Write B(X) for the space of Borel measurable functions on X, and B(X, R), B+(X) for the spaces of extended real-valued Borel measurable functions on X, and those that take values in [0, oo] respectively. We consider various distinguished subspaces of B(X) including C(X), Cb(X), C0(X) and CC(X) consisting of continuous complex-valued functions on X, those that are bounded, those that vanish at infinity, and those with compact support respectively. The positive cones of these vector spaces will be distinguished by using a superscript + sign. Both Cb(X) and C0(X) will be topologized by the uniform norm || · H^ whereas CC(X) will

1.1 Definition and general constructions

7

be topologized as the inductive limit of the spaces CE(X) :- {f e CC(X): supp(/) c £} with E compact, each of which carries the uniform norm. With these topologies all of the above spaces are complete. Another topology used in this book is that of compact convergence which we designate by tco. For any /e B(X) write pos(/) := {xeX:f(x)>0}. Michael topology. Let ^(X) denote the space of nonvoid compact subsets of X. For A, B c X write VA(B) := (C e V(X): C η Α φ 0 and C c B}. Then x~ of /C onto itself with the property (x~)~ = x for all x e K) such that (ε^ * £j,)~ = ε^- * ε χ - for all x, >> 6 Κ where μ~ denotes the image of μ under involution. HG7 For x, y e K, e e 5υρρ(εχ * ε^) if and only if x = y~. Usually in the literature, and in this book as well, it is the space (K, *) that is referred to as a hypergroup, and this is often written just as K. However it is important to realize that in general hypergroups have no algebraic structure of their own; all properties are inherited through the measure algebra. As such a study of the

10

1. Hypergroups and their measure algebras

harmonic analysis of hypergroups is in fact a study of the harmonic analysis of certain measure algebras. A familiar example of a hypergroup is a locally compact Hausdorff group G with 6 M (G) carrying its usual convolution structure. Many of the basic constructions are to be found in Dunkl (1973), Jewett (1975), Ross (1977) and Bloom and Heyer (19822); some of the more important ones will be presented in Section 1.1.4 below. It should be observed that the mapping μ -> μ~ is weakly continuous, positive and linear on Mb(K). We write μ~ for the adjoint of μ 6 Mb(K), which is defined by

μ~(Α):=μ-(Α)ίθΐ&\ΙΑε.

A hypergroup K will be called commutative if (Mb(K), +,*) is a commutative algebra. A hypergroup for which the involution is the identity mapping is called hermitian. It is easy to see from eχ jfc**" ρ Μ &

Λ

_

— IP Α Ρ ^ — \ ν°ΐ Λ ^"W/ /

ν

— P sie P — P sie P — "υ~ "r" — & υy ^γΛ _y Α

that every hermitian hypergroup is commutative. The positivity and continuity of the convolution mapping together with uniqueness of such mappings guarantee that for each μ, ν € Mb(K)

where the above equality is to be interpreted as holding by evaluating each of the measures μ * ν, εχ * ey at / for all / β CC(K). In view of this it is clear that the convolutions εχ * ey, x, y Ε Κ determine the entire hypergroup structure. In particular when considering examples we need only check associativity and commutativity for the convolution of point measures. It is easy to show that for μ, ν e Mb(K) we have \μ*ν\ε χ . It is worthwhile observing that any homomorphism τ satisfies τ(εχ * cy) = ετ(χ} * θτ00 for all x, y e X 1 . Furthermore anticipating the classification of idempotent measures we can show that τ(βΚί) = eK2 where eK. is the neutral element of X,, i — 1,2. Indeed the equality in the preceding sentence shows that ετ(βλ )*ε τ ( Βκ > = ετ(β(£ >, and then Theorem 1.6.7 below gives that ετ((,κ > = ωΗ for some compact subhypergroup of K 2 from which the claim follows.

1.1 Definition and general constructions

11

1.1.4 Constructions. In this section we present some of the important example types of hypergroups, and during the ensuing discussion the harmonic analysis and probability theory of hypergroups will be illustrated by reference to these. Later in Chapter 3 we shall develop in detail hypergroup structures on Z + and IR+. 1.1.5. Every locally compact group G is a hypergroup with its usual convolution structure. Aside from some small hypergroups the nonabelian locally compact groups provide the major source of known examples of noncommutative hypergroups. 1.1.6. A continuous action of a topological group H (with neutral element e) on a locally compact space X is a continuous mapping (x, s) -»x s from Χ χ Η to X such that xe = χ and (xs)' = xst for χ e X, s, t e H. We denote the orbit of χ e X under Η by X H = {xs: s € H], and the set of orbits by X" = {χ": χ e X}. Let G be a group. A mapping a: G -* G is called affine if there exist c e G and an automorphism φ of G such that a(x) = οφ(χ) for all χ e G. For topological groups G, Η a continuous affine action of H on G is a continuous action (x, s)-> x s for which each mapping x -»· xs is affine. If (x, s) -* x s is a continuous action of a compact group H on a nonvoid locally compact Hausdorff space X then XH is a decomposition of X into compact subsets, and XH is a closed subset of ^(X). The quotient topology on XH and the relative topology on XH are equal making XH into a locally compact Hausdorff space. The natural projection x -»x" is a continuous open mapping from X onto X ff . 1.1.7 Theorem. Let G be a locally compact group, H a compact group, and suppose that (x, s) -»· xs is a continuous affine action of H on G. Let c% be the normalized Haar measure on H, and give GH the quotient topology. For x, y e G define

Then (GH, *) satisfies properties HG1-HG4. Proof. We first observe that the convolution is well defined. Indeed if x" = x'H and yH — y'H then x' = xs and y' = y' for some s', t' € H, and using the invariance of e%

Γ Γ JH JH =

εχΗ

e{x,'V-t}H(oH(ds)toH

* S.yH

.

Now for each x e G define σχ e M/(G) by

I.

G

fdax =

JH

f(xs)· G such that x5 = c>,(x). Then as = (e*)'1 = (cXe))"1 = c~le~l = c~l gives fcf = e. Now (xy)s = cs4>s(xy) = cs(t>s(x)(/>s(y) = Define/eM^G) by λ =

JG

JH

f(a,)a>H(ds)

for all/ e CC(G). Then from (xsy')u = xs"asy'u we see that σχ*λ* ay is invariant under H,and

Hence the convolution on M b (G // ) is given by

(μ * ν)' = μ' * /. * ν' and this takes care of HG1 -HG4.

Π

1.1.8. Now let G be a locally compact group, and Η a compact subgroup with normalized Haar measure ωΗ. We give the set G//H := {HxH: x e G} of double cosets of Η the quotient topology. 1.1.9 Theorem.

The space G//H with '-= \ JH

is a hypergroup with identity Η = HeH and involution given by (HxH)~ = Proof. The mapping (x, (s, i)) -» s~lxt is a continuous affine action of Η χ Η on G with orbit space G//H. Hence by Theorem 1.1.7, (G//H,*) satisfies properties HG1HG4. That HeH is the identity follows readily from the choice of ωΗ as the normalized Haar measure on H, and the involution property is immediate. Π We devote Section 1.5.13 below to developing some of the harmonic analysis of double coset hypergroups.

1.1 Definition and general constructions

13

1.1.10. Let (x,s) -»· xs be a continuous action of a compact group H on a locally compact group G and suppose that each mapping χ -»· x s is an automorphism of G (this is a particular case of a continuous affine action). We give G" the quotient topology and write COH for the normalized Haar measure on H. 1.1.11 Theorem.

TVze space GH with

is α hypergroup with identity e" = {e} and involution given by (XH)

= (x l ) H .

Proof. Since the action is continuous and affine we can apply Theorem 1.1.7. We note that (xs/)H = (xst ly)'H = (xst~ly)" = (xy"~l)sH = and since ωΗ is the normalized Haar measure on H the convolution is just that given above. Q We observe that Theorem 1.1.11 includes Theorem 1.1.9 as a special case. 1.1.12. Let G be a compact group and GG := {x°: χ e G} its space of conjugacy classes with the quotient topology where x c := {t~1xt: t e G}. 1.1.13 Theorem.

The space GG with

is a compact commutative hypergroup with identity {e} and involution given by

(XG)- =(x-l)G.

Proof. The mapping (x,s) ->· s-1xs is a continuous action of G on G, and Theorem 1.1.11 applies. Π 1.1.14. Let G be a compact group. Its dual Σ consists of (equivalence classes of) continuous irreducible unitary representations of G. For U, V G Σ their tensor product U ® V can be decomposed into its irreducible components Ul,U2,..-,Un with respective multiplicities ml,m2,...,mn (see Hewitt and Ross (1970), Definition 27.29). We define a convolution on Μύ(Σ) by * ty ·—

iti d(U)d(V]Ku'

14

1. Hypergroups and their measure algebras

where d(U) denotes the dimension of U. Then Σ with this convolution and the discrete topology is a commutative hypergroup. Note that GG is by Theorem 1.1.13 a compact commutative hypergroup, and in fact it is isomorphic with the double coset hypergroup (G © G)//d(G) where o(G) := {(χ,χ): χ e G}. As we shall see in Chapter 2 below it has a dual (G G ) A identified with Σ. 1.1.15. As a particular example of 1.1.14 consider G = SU(2) in which case Σ can be identified with Z+ as follows. By Hewitt and Ross (1970), Example 29.13 the set of continuous unitary irreducible representations of SU(2) is given by {T(0), Γ (1) , T (2> , . . . } where T(n) has dimension η + 1; we identify this set with Z+. For every m, n e Z + the tensor product T(m) ® T(n) is unitarily equivalent to

(see Hewitt and Ross (1970), Theorem 29.26 for details). The convolution is given by (k+l)

where the prime denotes that only every second term appears in the sum. With this convolution Z + becomes a commutative (discrete) hypergroup, and since all the T(M) are self-conjugate the hypergroup is in fact hermitian. 1.1.16. Let S be the unit sphere in (R3, and G := S0(3) the (special orthogonal) group of all orientation-preserving isometries of S. Write N P := (Ο, Ο, 1) and H := {g 6 G: G(NP) = NP}. Then H is a closed subgroup of G isomorphic to the circle group. Two elements of G are conjugate if they rotate S through the same angle, which we represent by a number in [Ο, π]. We apply Theorem 1.1.13 to obtain a compact commutative hypergroup structure on [Ο, π] with identity 0 and involution the identity map. The support of εχ * ey is either an interval or a singleton; in particular supp(e„ * εη) = [Ο, π]. 1.1.17. We now consider G//H with G, H as in 1.1.16. For each χ 6 [— 1, 1] write Dx = {g 6 G: g(NP) = (s, t, x) for some s, t}. Each set Dx is a double coset of H, and Theorem 1.1.9 applies to give a compact commutative hypergroup structure on [—1,1] with identity 1. In fact since the only topological involution of [—1,1] leaving 1 fixed is the identity mapping this hypergroup is hermitian and hence commutative. Note that ε_! * ε_! = £j so that this hypergroup is not isomorphic to that given in 1.1.16. 1.1.18. Let G = IR2 and Η = J. There is a natural continuous action of Η on G with each mapping being a rotation. These rotations are automorphisms, and the orbits are concentric circles with centre (0,0). We use K+ as a model of Gu with each positive number χ representing the circle of radius x. The convolution operation can

1.2 Translation and convolution

15

be written as

-ΙΓ

Jlx-j

for x, y e R + . Then 0 is the identity, and the hypergroup is hermitian.

1.2 Translation and convolution In the definition of a hypergroup convolution is given initially for point measures. With the continuity and positivity assumptions on the convolution mapping this operation is easily extended to all complex measures. It remains to investigate the convolvability of other measures, of functions and measures, and in fact of functions and functions. Preliminary to this we introduce the concept of translation in a hypergroup. 1.2.1 Translation of functions. For / e B(K), x, ye K f(x*y):=

fd(Ex*sy) ν «

if this integral exists (though it need not be finite). The above notation is suggestive of f(xy) which is the case when K is a locally compact semigroup. However note that x * y has no meaning on its own. We observe immediately that /~(x * y) = f(y~ * x~). It is convenient at this stage to write Txf(y) = f(x * y) for the left x-translate of/ at y, and Txf(y) = f(y * x) for the right x-translate of/ at y. We can extend these two definitions to μ e Μ (K) by defining (Γ*μ)(/) = μ(Τχ/) and (7»(/) = μ(Τχβ for all / e CC(K), x e K. It is easy to show that Txf e CC(K) whenever / e CC(K); see Proposition 1.2.16(iii) below. Properties of these translation operators Tx, Tx will be investigated in the following sections. At this stage we note by HG3 that for / e Cb(K) the mapping (x, y) -»/(x * y) is a continuous function on Κ χ K, and hence each Txf, Txf is a continuous function on K. Every / e C(K) satisfies the following local uniform continuity property. 1.2.2 Theorem. Let f e C(K) and let C be a compact subset of K. Then for x0 e K and ε > 0 there exists a compact neighbourhood V of x0 such that for all x e V, y e C

\f(y~*x)-f(y~*xo)\

) < ε/(2||/ !!,„). Then using Lemma 1.2.11, (D* K\(C*£>))n C = 0, and for x ^ C*Z> < ί |/(^*χ)||μ|(^)+ ί Jfl

JK\

=ε

which shows that μ*/ε C0(X). The second assertion follows by approximating by functions and measures with compact support. Π As would be expected all translation and convolution operators are continuous. The continuity of the translation operator with respect to the topologies τυ and rw can be proved as for the group case (see Berg and Forst (1975), p. 7). 1.2.17 Theorem. The following mappings with the topologies as indicated are continuous. (i) (χ,μ) -» Γ*μ of Κ χ M+(K) into M+(K); (Μ+(Κ),τ0). (ii) (χ,μ) -» Τχμ of Κ χ Mb+(K) into Mb+(K); (Mb+(K),iw). Proof, (i) Suppose x, ->· χ in Κ and T W — lim μ, — μ in M+(K). We want to show that Τχ>μ, -»· Τχμ or equivalently (since by Proposition 1.2.16(iii), T*f e CC(K) for all /eC c (K))

f r*/^, -> [

JK

JJ

Choose C/ a compact neighbourhood of x, and then i'0 such that x,€ U for all / > ioPut D := U~ * supp(/). Then for i > / 0 supp(r*/) = {x,-} * supp(/) c (7- * supp(/) = D and since x e U the same is true for suppCT*/).

1.2 Translation and convolution

21

Now consider (for ι > ι'0)

Τ^άμ, IK

Τ' JK

and let ε > 0. To estimate the first term use Theorem 1.2.2 to choose ι'ό such that for all ι > ig I Tx'f(y) - T*f(y)\ < ε for all y e D. Then

-L

< εμ,(Ο)

for all t>%.

To estimate the second term just note that τν — lim μ, = μ implies

asTxfeCc(K). Finally we observe that limsup/i^/)) < μ(Ο) which can be proved by majorizing 1D by suitable g e C*(K). Now choose ;0 > ι'0 ν ι'ό suitably large so that all of the above estimates hold, and the result is proved. (ii) The proof of this part (which is similar to that of (i)) is straightforward. Π 1.2.18 Properties. (1.2.19)

For μ, ν e Mb(K) and / e B(K), μ * / e B(K) and μ-

_ Γ

_Γ

JA

JK

Furthermore μ * (ν */) = (μ* ν)*/, μ*(/* ν) = (μ*/)* ν and (μ*/) = / *μ .

1.2.20. Let (/„) be an increasing sequence in B+(K) converging to /. Then it follows from the monotone convergence theorem that for x, y e Κ and μ e Mb(K), fn(x * y) -> f(x * y) and μ */„ -> μ */. 1.2.21 Lemma. For μ, ν e Mb(K), A e (μ*ν)(Α)=

JK

μ *\Adv.

22 Proof.

1. Hypergroups and their measure algebras (μ*ν}(Α}=\

\A(s*t^(ds)v(di) JK JK

•UI -iJ>

JK JK JK

JK JK

JK

-l·-·· JK

where the last equality follows from

'(«W

= (μ*εχ)(Α) and this completes the proof.

Π

We can prove a range of "regularity" properties. 1.2.22 Lemma. Let f, k e C+(K) with k φ 0. There exists μ e MCr+(K) such that

Proof. If fc(ii) > 0 then (εα- */c)(